\1cw Verze 2.10 \pTM 2 \pBM 2 \pPL 124 \pLM 1 \pRM 65 \HD \+ \+ \, \- \= \HE \+ \+ \, \- \= \FD \+ \+ \, \- \= \FE \+ \+ \, \- \= \+ \+ \"5. P A R T I T I O N S\, \-\1 \+ \+ \45.1 Preliminary notes\, \-\1 \+ \+\ \ \ Partitions of a natural number m\ into n parts were introduced \-\/ \+ \+ into mathematics by Euler. There exists an analytical formula for \- \+ \+ finding the number of partitions, \ derived by Ramanudjan and Har-\A \- \+ \+ dy. Ramanudjan was a mathematical genius \ from India. He was sure \- \+ \+ that it is possible to calculate the number of partitions exactly \- \+ \+ for any number m. He found \ the solution in cooperation with Eng-\A \-\/ \+ \+ lish mathematician Hardy. It \ is rather complicated formula deri-\A \- \+ \+ ved by \ higher mathematical techniques. \ We will use \ only simple \-\/ \+ \+ recursive methods.\, \- \+ \+ A partition splits a number into parts \ which sum is eqal to \-\/ \+ \+ the number, say 7: 3,2,1,1, is an ordered set. Its part are writ-\A \- \+ \+ ten in \ decreasing order. If you \ look on a string of \ parts, you \- \+ \+ see \ that it \ is a vector \ row. From \ a partition vector, another \- \+ \+ vectors having \ equivalent structure of elements \ are obtained by \- \+ \+ permuting, simple changing of ordering of vector elements. We ne-\A \-\/ \+ \+ ed partitions for ordering of plane simplices points.\, \- \+ \+ All unit permutations of a vector have the same length. The-\A \-\/ \+ \+ refore \ different partitions \ form bases \ for different \ vectors. \- \+ \+ Vectors belonging to the same \ partition are connected with other \- \+ \+ points of the \ simplex by circles. In higher \ dimensions the cir-\A \- \+ \+ cles become \ spheres and therefore \ we will call \ a partition \4the \-\1\/ \+ \+ \4partition orbit \1or simply orbit.\, \- \+ \+ The number of nonzero vectors in partitions will be given as \-\2\/ \+\1 \+ n, the size of the first vector as m . Zeroes will be not written \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \+\1 \+ to spare work. The bracket (m,n) means all partitions of the num-\A \- \+ \+ ber \ m into \ at \ most \ n parts. \ Because \ we write a partition as \, \- \+ \+ a vector, we allow zero parts in a partition. It is a certain in-\, \- \+ \+ novation against the tradition, which will be very useful.\, \- \+ \+ \4\ 5.2 Ferrers graphs\, \-\1 \+ \+ Ferrers graphs are used in the theory of partitions for many \-\/ \+ \+\2 \1proofs. They are tables containing \ m objects, one object in each \-\2\/ \+\1 \+\2 \1compartment, \ arranged \ into \ columns \ in \ nonincreasing order m \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n \9> \1m with \ the sum \7S \1m = m. \ It is obvious \ that a Ferrers \-\2\ \ \ j+1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j=1\ \ j \+\1 \+ graph is a matrix which has its unit elements arranged consecuti-\A \- \+ \+ vely in initial rows and columns. When compared with naive matri-\A \- \+ \+ ces, Ferrers graphs look like \ squeezed naive matrices \4N\1, which \- \+ \+ all unit elements \ were compressed to the row \ baseline (it is up \-\/ \+ \+ in matrices) by eliminating empty elements.\, \- \+ \+ Introducing Ferrers graphs as matrices, we come to the notion of \-\/ \+ \+ restricted partitions. \ The parts of a partitions \ can not be gre-\A \- \+ \+ ater than the number of rows of the matrix, and the number of parts \- \+ \+ greater than \ the number of \ columns. Now we need \ to sophisticate \- \+ \+ the notation to distinguish the partitioned number M and the num-\A \- \+ \+ ber of rows m. The unrestricted number of partitions p(M) is equal \- \+ \+ to the number of \ restricted partitions, if restricting conditions \-\/ \+ \+ are loose, m \ \9> \1M and n \9> \ \1M: p(M) = p(M,M,M). \ We write here at \- \+ \+ first the \ number of rows m, \ then the number of \ parts n, and at \- \+ \+ last the sum of unit elements M.\, \- \+ \+ If we transpose Ferrers graphs, we get an important property of \- \+ \+ restricted partitions :\, \- \+ \+ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p(m,n,M) = p(n,m,M) (5.1)\, \- \+ \+ The number of partitions into exactly n parts with the grea-\A \-\/ \+ \+\2 \1test part m is the same \ as the number of partitions into n parts \-\2\/ \+\1 \+\2 \1having the greatest part n.\, \-\2 \+\1 \+ We \ can subtract \ a Ferrers \ graph from \ the matrix containing \- \+ \+ unit elements only and translate the resulting matrix ,e.g.\, \-\2 \+\1 \+\8\ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ \ \ p\ \ \ \ \ p p \11 1\8p p\11 0\8p p\10 1\8p p\11 1\8p\, \-\ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ \ \ p\ \ \ \ \ p \+ p p \1- \8p p \"= \8p p -----\1>\ \8p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ \ \ p\ \ \ \ \ p p \11 1\8p p\10 0\8p p\11 1\8p p\11 0\8p\, \-\1 \+ \+ We get the relation between the number of restricted partitions \- \+ \+ of two different numbers\, \- \+ \+ \ \ \ \ \ \ \ \ \ \ \ \ \ p(m,n,M) = p(n,m,mn - M) (5.2)\, \- \+ \+ In the proof of this identity, we have used an operation ve-\A \-\/ \+ \+ ry usefull \ for deriving elements \ of partition schemes \ and res-\A \- \+ \+ tricted \ partitions \ of \ all \ kinds. \ A restricted partition into \- \+ \+ exactly n parts, having m as the greatest part has (m + n - 1) u-\A \- \+ \+ nits bounded by elements forming the \ first row and column of the \- \+ \+ corresponding Ferrers graphs (Fig.5.1). Only (M - m - n + 1) ele-\A \- \+ \+ ments are free \ for partitions in the restricted \ frame (m-1) and \-\/ \+ \+ (n-1). Therefore\, \- \+ \+ \ \ \ \ \ \ \ \ \ \ p(m,n,M) = p(m-1,n-1,M-m-n+1) (5.3)\, \- \+\7 \+\8 \1E.g.:p(4,3,8) = p(3,2,2) = 2. The corresponding \ \ partitions \ are \-\2\/ \+\7 \+\8 \14,3,1 and 4,2,2, \ or 2,0 and 1,1, respectively. \ This formula can \-\2 \+\8 \+\1 be used for finding all restricted partitions. It is rather easy, \-\8 \+\1 \+ if the difference (M-m-n+1) is smaller than the restricting valu-\A \-\8 \+\1 \+ es m,n, \ or at least \ one from them. \ The row and \ column sums of \-\8 \+\1 \+ partially restricted partitions having \ the other constrain cons-\A \-\8\/ \+\1 \+ tant, where either n or m can be 1 till M are:\, \-\8 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M \1\ \ \ \ \ \ \ \ \ \ \ p(m,*,M) = \7S\ \ \1p(m,j,M) (5.4)\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j=1 \+\8 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M \1\ \ \ \ \ \ \ \ \ \ \ p(*,n,M) = \7S\ \ \1p(i,n,M) (5.5)\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1 \+\1 \+ Partition schemes are due to the transpositions symmetrical, \-\/ \+ \+ the row and \ column sums are equal. Before \ we examine restricted \- \+ \+ partitions in more detail, we will introduce tables of unrestric-\A \-\/ \+ \+ ted and partially restricted partitions.\, \- \+ \+ \45.3 Partition matrices\, \-\1 \+ \+ Partially restricted partitions can \ be obtained from unres-\A \- \+ \+ tricted partitions by subtracting a row of n units or a column of \- \+ \+ m units, respectively. \ This gives us \ the recursive formula \ for \-\/ \+ \+ the number of partitions as a sum of two partitions \, \- \+ \+ p(*,N,M) = p(*,N-1,M-1) + p(*,N,M-1) (5.6)\, \- \+ \+ We divide all partitions into exactly N parts into two sets. In \-\/ \+ \+ one set are partitions having in \ the last column 1, their number \- \+ \+ is counted by the term p(*,N-1,M-1) which is the number of parti-\A \- \+ \+ tions of the number (M-1) into exactly (N-1) parts to which 1 was \- \+ \+ added on the nth place and in other set are partitions which have \- \+ \+ in the last \ column 2 and more. They were obtained \ by adding 1 to \-\/ \+ \+ all n elements of the partitions of (M-N) into N parts.\, \- \+ \+ A similar formula can be deduced for partitions of M into at \-\/ \+ \+ most N parts. These partitions can have zero at least in the last \-\/ \+ \+ column or they are partitioned into n parts exactly :\, \- \+ \+ p(*,*=N,M) = p(*,*=N-1,M) + p(*,*=N,M-N) (5.7)\, \- \+ \+ To formulate both recursive formulas more precisely, we must \-\/ \+ \+ define an apparently paradoxical partition: p(0,0,0) = 1. What it \- \+ \+ means? \ A partition of zero \ into zero number of \ \ parts. \ \ This \-\/ \+ \+\2 \1partition represents the empty space of dimension zero. This par-\A \-\2 \+\1 \+\2 \1tition can be justified as a limit. Using our generating function \-\2 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ m\ \ \ \ 0 \1we \ write n = 0 and \ find its \ limit: \, \-\2\/ \+\1 \+\2\ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ 0 \1lim \ 0 = lim (1/x) = 1/x = 1.\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ x-->\98 \+\1 \+ We get two following tables of partitions\, \- \+ \+ \4Table 5.1 Partitions into exactly n parts\, \-\1 \+\8----i----------------------------i--- \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n \8p \10 1 2 3 4 5 6 \8p \7S\, \-\1 \+\8----k----------------------------k---- \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=0 \8p \11 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \11\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 1 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \11\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p \11 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \12\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p \11 1 1 \8p \13\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \11 2 1 1 \8p \15\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \11 2 2 1 1 \8p \17\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p \11 3 3 2 1 1 \8p \111\, \- \+ \+ \, \- \+ \+ \4Table 5.2 Partitions into at most n parts\, \-\1 \+\8\ --i--------------------------- \+\ \ \ p \1n \ 0 1 2 3 4 5 6\, \-\8---k--------------------------- \+\ \ \ p \+\ \ \ p \1m=0\8p \11 1 1 1 1 1 1\, \-\8\ \ \ p \+\ \ \ p \+\ \ \ p \1 1\8p \11 1 1 1 1 1\, \-\8\ \ \ p \+\ \ \ p \+\ \ \ p \1 2\8p \11 2 2 2 2 2\, \-\8\ \ \ p \+\ \ \ p \+\ \ \ p \1 3\8p \11 2 3 3 3 3\, \-\8\ \ \ p \+\ \ \ p \+\ \ \ p \1 4\8p \11 3 4 5 5 5\, \-\8\ \ \ p \+\ \ \ p \+\ \ \ p \1 5\8p \11 3 5 6 7 7\, \-\8\ \ \ p \+\ \ \ p \+\ \ \ p \1 6\8p \11 4 7 9 10 11\, \- \+ \+ Table 5.2 is obtained from the \ Table 5.1 as partial sums in \-\2\/ \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T \1rows, it means by multiplying \ with the unit triangular matrix \4T \-\2 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T \1from the \ right. The elements of \ the matrix \4T \1are \ h = 1 if j \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ij \+\1 \+\2 \9> \1i, h = 0 if \ j < i. Otherwise, the Table \ 5.1 is obtained from \-\2\ \ \ \ \ \ ij \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -T \1the Table \ 5.2 by multiplying \ with a matrix \4T \ \1from the right. \-\2 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1\ \ \ \ \ \ \ -1 \1The inverse elements are h = 1, h = -1, h = 0 otherwise. \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ii\ \ \ \ \ \ \ i,i+1\ \ \ \ \ \ \ \ ij \+\1 \+ Notice, that the \ diagonal elements of the Table \ 5.2 remain con-\A \- \+ \+ stant \ at higher \ n. Increasing \ the numbers \ of zeroes \ does not \-\/ \+ \+ change the number of partitions.\, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -T \1 When we multiply Table 5.1 by the matrix \4T \1again, we obta-\A \-\/ \+ \+ in partitions having \ as the smallest allowed part \ the number 2. \- \+ \+ The effect of \ these operators can be visualized \ on the 2 dimen-\A \- \+ \+\2 \1sional complex, the operators shift \ the border of counted orbits \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -T \1(Fig.5.2). The operator \ \4T \1differentiates n dimensional comple-\A \- \+ \+ xes, shifting their border to \ positive numbers and cutting lover \-\/ \+ \+ numbers. Zero seems to form the base border.\, \- \+ \+ \45.4 Partitions with negative parts\, \-\1 \+ \+ Operations with \ tables of partitions lead \ us to a thought, \-\/ \+ \+ that it were possible to go outside the positive cone to nonnega-\A \-\/ \+ \+ tive numbers and allow the \ existence of negative numbers in par-\A \-\/ \+ \+ titions.\, \- \+ \+ If we write the number of equal parts n as the vector row un-\A \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\/ \+\1 \+ der the vector formed by the number scale, we see that the number \- \+ \+ of partitions \ is independent on \ shifts of the \ number scale, see \- \+ \+ Table 5.3. \ Partitions are derived \ by shifting allways \ two vec-\A \- \+ \+ tors, one 1 position up, the other 1 position down. Each partition \- \+ \+\2 \1corresponds to a vector. If we write them as columns, then their \-\2 \+\1 \+\2 \1scalar \ product with \ the number \ scale, forming \ the vector \ row \-\2 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T \4m \1gives constant sum : \4mp \1= \7S \1m n \ = m. There is an inconsis-\A \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\9>\2r k\ k \+\1 \+\2 \1tency, elements of the vector \4p \1are numbers of vectors having the \-\2 \+\1 \+\2 \1same length and we use for them the letter n with an index k. For \-\2 \+\1 \+\2 \1values of the \ number scale we use the letter \ m, with the common \- \+ \+\2 \1index k, which goes from the \ lowest allowed value \ of parts till \- \+ \+\2 \1the highest possible value. We can continue till infinity, but all \-\/ \+ \+ following values n will be zeroes.\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k \+\1 \+ \4Table 5.3 Partitions as vectors\, \-\1 \+ \+\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1Parameter r\ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\1 \8----------i---k--------------------i-----------\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \2T \1Vector \4m \8p \1-2\8p \1-1 0 1 2 3 \8p \4mp\ \1= -5\, \-\8\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \1-1\8p \10 1 2 3 4 \8p \10\, \-\8\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \10\8p \11 2 3 4 5 \8p \15\, \-\8\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \11\8p \12 3 4 5 6 \8p \110\, \-\8\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \12\8p \13 4 5 6 7 \8p \115\, \-\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+----------k------------------------k----------- \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1Vector n \8p \14 1 \8p\ \, \-\2\ \ \ \ \ \ \ \ k\ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \13 1 1\ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \13 1 1\ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \12 2 1\ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \12 1 2\ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \11 3 1\ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \11 2 2\ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 \ 5\, \-\8\ \ \ \ \ \ \ \ \ \ m------------------------. \+\1 \+ Inserting for different vectors m and n we get:\, \- \+ \+ 4*-2 + 1*3 = -5,\, \- \+ \+ 3*-1 + 1*0 + 1*3 = 0\, \- \+ \+ 3*0 + 1*2 + 1*3 = 5\, \- \+ \+ 2*1 + 1*2 + 2*3 = 10\, \- \+ \+ 1*2 + 3*3 = 1*4 = 15.\, \- \+ \+ The parameter r shifts the table \ of partitions, its front ro-\A \-\/ \+ \+ tates around the zero point. If \ r were -\98\1, then p(-\98\1,1) = 1, but \- \+ \+ p(-\98\1,2) were undetermined, because \ a sum of a finite number with \- \+ \+ an infinite number \ is infinite. We can write \ the parameter r to \- \+ \+ a partition as its upper index, to show that different bases of \-\/ \+ \+ partitions are differentiating plane simplices.\, \- \+ \+ \45.5 Partitions with inner restrictions\, \-\1 \+ \+ We have \ classified partitions according to \ the minimal and \-\/ \+ \+ maximal allowed values of parts, but there can be restrictions in-\A \- \+ \+ side the number scale, we can \ prescribe that some values should be \- \+ \+ forbidden. It is easy to see what it means: The plane simplex has \- \+ \+ holes, some \ orbits can not \ be realized and \ its (n -1) \ body is \-\/ \+ \+ thinner than the normal one.\, \- \+ \+ It is \ easy to find the \ number of partitions which \ all parts \-\/ \+ \+ must be even. \ It is not possible to form \ an even partition from \-\2 \+\1 \+ an uneven \ number, therefore: p (2n) = p (n). More \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ even \ \ \ \ \ \ \ \ unrestricted \+\1 \+ difficult a task \ is to find \ the number of \ partitions which all \- \+ \+ parts \ must be \ odd. Other partitions \ contain mixed \ odd and even \-\/ \+ \+ parts. The relation between different partitions is\, \- \+ \+ p(unrestricted - odd - even) =p(mixed). \, \- \+ \+ Their list is in Table 5.4\, \- \+ \+ \4Table 5.4 Odd, even, and mixed partitions\, \-\1 \+\8---i-----------------------------------i-----------------------o \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \ \1Number of odd partitions \8p \1Sums\ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+---k-----------------------------------k----i-----i------i-----o \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1n \8p \11 2 3 4 5 6 7 8 9 \8p \1Odd\8p \1Even\8p \1Mixed\8p \1p(m)\8p\, \-\1 \+\8---k-----------------------------------k----k-----k------k-----l \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1m=1\8p \11 \8p \11 \8p \10 \8p \10 \8p \11\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 2\8p \11 \8p \11 \8p \11 \8p \10 \8p \12\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 3\8p \11 1 \8p \12 \8p \10 \8p \11 \8p \13\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 4\8p \11 1 \8p \12 \8p \12 \8p \11 \8p \15\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 5\8p \11 1 1 \8p \13 \8p \10 \8p \14 \8p \17\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 6\8p \12 1 1 \8p \14 \8p \13 \8p \14 \8p \111\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 7\8p \11 2 1 1 \8p \15 \8p \10 \8p \110 \8p \115\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 8\8p \12 2 1 1 \8p \16 \8p \15 \8p \111 \8p \122\ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ p \1 9\8p \11 3 2 1 1 \8p \18 \8p \10 \8p \122 \8p \130\ \ \8p\, \-\1 \+ \+ Notice, how the scarce matrix of odd partitions is made from \-\/ \+ \+ Table 5.2. Its elements, except the first one in each column, are \- \+ \+ shifted down on cross diagonals. \ An odd number must be partitio-\A \- \+ \+ ned into an odd number of odd \ parts and an even number into even \- \+ \+ number of odd parts. Therefore, the \ matrix can be filled only in \- \+ \+ half. The reccurence is given by two possibilities, how to incre-\A \- \+ \+ ase the number m. Either we add odd 1 to odd partitions of (m-1) \- \+ \+ with exactly \ (j-1) parts or we \ add 2j to odd \ numbers of parti-\A \- \+ \+ tions of \ (m-2j) with exactly j parts. \ The relation is expressed \-\/ \+ \+ as o(i,j) = p[(i+j)/2,j].\, \- \+ \+ Partitions with all parts unequal are important, their trans-\A \-\/ \+ \+ posed Ferrers graphs have the greatest part odd, if the number of \-\/ \+ \+ parts is odd, and even, if the number of parts is even E.g. \, \- \+ \+ \ \ \ \ \ \ \ \ \ \ 10\, \- \+ \+ \ \ \ 9,1\, \- \+ \+ 8,2\, \- \+ \+ 7,3 7,2,1\, \- \+ \+ 6,3 6,3,1 \, \- \+ \+ 5,4,1 \, \- \+ \+ 5,3,2\, \- \+ \+ 4,3,2,1\, \- \+ \+ The \ partitions with \ unequal parts \ can be \ tabulated as in \-\/ \+ \+ Table 5.5. Notice that the difference of the even and odd columns \- \+ \+ partitions is mostly zero and \ only sometimes (+/- 1). The impor-\A \- \+ \+ tance of this \ phenomenon will be explained later. \ The number of \-\/ \+ \+ partitions \ with unequal \ parts coincidence \ with the \ partitions \-\/ \+ \+ which all parts are odd.\, \- \+ \+ \4Table 5.5 Partitions with unequal parts\, \-\1 \+\8-----i--------------------i-----i------------------------- \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1n \8p \11 2 3 4 \8p \1Sum \8p\1Difference (n odd-n even)\, \- \+\8-----k--------------------k-----k-------------------------- \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1m=1 \8p \11 \8p \11 \8p \11\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 2 \8p \11 \8p \11 \8p \11\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 3 \8p \11 1 \8p \12 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 4 \8p \11 1 \8p \12 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 5 \8p \11 2 \8p \13 \8p \1-1\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 6 \8p \11 2 1 \8p \14 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 7 \8p \11 3 1 \8p \15 \8p \1-1 \, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 8 \8p \11 3 2 \8p \16 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 9 \8p \11 4 3 \8p \18 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 10 \8p \11 4 4 1 \8p \110 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 11 \8p \11 5 5 1 \8p \112 \8p \10\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p \1 12 \8p \11 5 7 2 \8p \115 \8p \11\, \-\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\1 \+ The \ differences \ are \ due \ to Franklin blocks \ with growing \-\/ \+ \+ minimal parts and growing number \ of parts (we use their transpo-\A \- \+ \+ sed notation), \ which are minimal \ in the sense, \ that their part \-\/ \+ \+ differs by one, the shape is trapeze:\, \- \+ \+ 1 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1, 2\, \-\8-------------------------------------- \+\1 \+ 1 1 1 1 1 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ 5, 7\, \- \+ \+ 1 1 1 1 1\, \-\8------------------------------------------- \+\1 \+ 1 1 1 1 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 1 1 1 1 1\ \ \ \ \ \ \ \ \ \ 12, 15\, \- \+ \+ 1 1 1 1 1 1 1 1 1\, \- \+ \+ 1 1 1 1 1 \- \+ \+ \45.6 Differences according to unit parts\, \-\1 \+ \+ We have arranged restricted \ partitions according the number \- \+ \+ of zero parts in Table 5.3. It is possible to classify partitions \- \+ \+ according the number of vectors with any value. Using value 1, we \-\/ \+ \+ get another kind of partition differences as in Table 5.6 \, \- \+ \+ \4Table 5.6 Partitions according to unit parts\, \-\1 \+\8----i---------------------------o \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n\ \8p \10 1 2 3 4 5 6\8p\, \-\2\ 1 \+\8----k---------------------------l \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=0 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 1 \8p \10 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p \11 0 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p \11 1 0 1\ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \12 1 1 0 1\ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \12 2 1 1 0 1\ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p \14 2 2 1 1 0 1\8p\, \-\1 \+ \+ The elements of the table are: p\ \ = p(i) - p(i - 1),\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i0 \+\1 \+ pöö = \ pöööööööö otherwise. \ Table \ 5.6 \ is \ \ obtained \ from \ the \-\2\ ij\ \ \ \ \ \ i-1,j-1\/ \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1 \1folloving Table 5.7 by multiplying with the \ matrix \4T \2. \1The zero \-\2\/ \+\1 \+ column is \ the difference of two \ consecutive unrestricted parti-\A \-\2 \+\1 \+ tions according \ to m. To all \ partitions of p(m-k) we \ added k*1 \-\2 \+\1 \+ and partitions \ in the zero \ column contain only \ numbers greater \-\2 \+\1 \+ than 1. These partitions can \ not be formed from lower partitions \-\2 \+\1 \+ by adding \ ones and they \ are thus a difference \ of the partition \-\2\/ \+\1 \+ function according to the number nö. Since Table 5.7 is composed, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \+\1 \+ its inverse is composed, too.\, \-\2 \+\1 \+ \45.7 Euler inverse of partitions\, \-\1 \+ \+ If we write successive partitions as column or row vectors as \-\2\/ \+\1 \+ in Table 5.7, which elements are \ p = p(i - j + 1), we can find \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ij \+\1 \+ rather \ easily its \ inverse matrix, \ which you \ find in the second \-\2\/ \+\1 \+ part of the same table.\, \-\2 \+\1 \+ \, \- \+ \+ \4Table 5.7 Partitions and their Euler inversion\, \-\1 \+\8u--i----------------------------i-----------------------o \+p\ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p p \1Partition table \8p \1Euler inversion \ \ \ \ \ \8p\, \-\1 \+\8m--k----------------------------k-----------------------l \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1j \ 0 1 2 3 4 5 0\ 1 2 3 4 5 \, \-\8---k----------------------------k-----------------------l \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1i=0\8p \11 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 1\8p \11 1 \8p \1-1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2\8p \12 1 1 \8p \1-1 -1 1\ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3\8p \13 2 1 1 \8p \10 -1 -1 1\ \ \ \ \ \ \ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4\8p \15 3 2 1 1 \8p \10 0 -1 -1 1\ \ \ \ \8p\, \-\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5\8p \17 5 3 2 1 1 \8p \11 0 0 -1 -1 1\8p\, \-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\1 \+ The nonzero elements in the \ first (and similarly in further \-\/ \+ \+ columns, which are only always \ more interesting shifted down one \- \+ \+ row) of \ the inverse matrix appear \ at indices, which can \ be ex-\A \-\/ \+ \+ pressed \ by the \ Euler \ identity \ concerning the \ coefficients of \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ 3 \1expansion of (1 - t)(1 -\ t )(1 - t )... =\, \- \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ \ \ \ 2 \+\9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8\ \ \ \ \ \2i\ \ (3i -i)/2 \ \ \ (3i\ +i)/2 \1\ \ \ \ \ \ \ \ \ \ \ 1 + \7S\ \ \1(-1)\ [t\ \ \ \ \ \ \ \ + t\ \ \ \ \ \ \ ] (5.8)\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1 \+\1 \+ For example for the last row: 7 -5 - 3 + 0 + 0 + 1 = 0. \, \- \+ \+ When i = 1, we have the pair of indexes at t 1, 2 with nega-\A \-\/ \+ \+ tive sign; for i = 2 the pair is \ 5, 7; for i = 3 the pair -12, - \- \+ \+ 15 and so on these numbers are distances from the base partition. \- \+ \+ The inverse matrix \ becomes scarcer as p(m) increase, \ as we have \- \+ \+ already shown as Franklin partitions, showed before. \ All inverse \-\/ \+ \+\9 \1elements are -1,0,1. The nonzero elements of the Euler polynomial \-\2 \+\1 \+\9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8\ \ \ \ \ \ \2i \1are obtained as sums of the product \7P \1(1 - t ). This can be ve-\A \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=1 \+\1 \+\9 \1rified by \ multiplying several terms of \ the infinite product. If \-\2 \+\1 \+\9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \1we \ multiply \ the \ Euler \ polynomial with \ its \ inverse \ function \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \+\1 \+\9\ 8\ \ \ \ \ \ \ \ \ \2i\ -1 \7P \1(1 - \ t ) \2, \1we \ obtain 1. \ From this \ relation follows, that \-\2\ i=1 \+\1 \+\2 \1partitions are \ generated by the inverse \ Euler function which is \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i \4the generating function \1of \ partitions. Terms t must be conside-\A \-\/ \+ \+\2 \1red as representing unequal parts.\, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i \1 The Euler function has all parts \ t different. We have con-\A \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i \1structed such partitions \ in Table 5.6. If the \ coefficient at t \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i \1is obtained as the product of even number of (1 - t ) terms, the \- \+ \+ sign is positive, and if it is the result of the uneven number of \- \+ \+ terms, \ it is \ negative. The \ coefficients are \ determined by the \- \+ \+ difference of the \ number of partitions with odd \ and even number \- \+ \+ of unequal parts. This difference can be further explained accor-\A \-\/ \+ \+ ding to Franklin using Ferrers graphs.\, \- \+ \+ All parts having as at least \ one part equal 1, are obtained \- \+ \+ from p(n-1). The difference is due to p(n-2). We must add to each \- \+ \+ partition 2, \ and all 1 must be \ removed. This can be \ done using \-\/ \+ \+ transposed \ Ferrers \ graphs, \ since \ \ we \ need \ big \ parts. \ Thus \- \+ \+ partitions are formed from conjugate partitions, too. Unused par-\A \-\/ \+ \+ titions must be subtracted. For example, p(8):\, \- \+ \+\2\ \ \ \ \ \ \ \ \ 6 \1 6; 1 ; Formed: 8; 62;\, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ 4 \1 51; 21 ;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 53;\, \-\ \ \ \ \ -- \+ \+\2\ \ \ \ \ \ \ \ \ \ 2\ 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4 \1 42; 2 1 ;\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 44; 2 ;\, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 \1 33; 2 ; 3 2;\, \-\ \ \ \ \ -- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 \1 41 ; 31 ; 42 ; \, \-\ \ \ \ \ \ \ \ \ \ -- \+ \+ 321;\, \-\ \ \ \ \ --- \+ \+ \ \ \ \ \ Leftovers (underlined above):\, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3 \1 p(1): 51; p(3): 33; 321; 31 \, \- \+ \+\2 \1are obtained \ by subtracting the largest \ part from corresponding \- \+ \+\2 \1partition. \ Two must \ be added \ \ to the \ subtracted part. \ We get \- \+ \+\2 \1p(8-5) and p(8-7) as the corrections.\, \- \+ \+ \45.8 Other inverse functions of partitions\, \-\1 \+ \+ We already met other tables \ of partitions which have inver-\A \-\/ \+ \+ ses, since they \ are in lower triangular form. The inverse to the \-\/ \+ \+ Table 5.1 is Table 5.8\, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \4Table 5.8 Inverse matrix to partitions into n parts\, \-\1 \+\8----i-----------------------o \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n \8p \11 2 3 4 5 6\8p\, \-\1 \+\8----k-----------------------l \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=1 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p \1-1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p \10 -1 1\ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \11 -1 -1 1\ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \10 1 -1 -1 1\ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p \10 1 0 -1 -1 1\8p\, \-\1 \+ \+ The inverse to Table 5.6 is Table 5.9\, \- \+ \+ \4Table 5.9 Inverse matrix of unit differences\, \-\1 \+\8----i-----------------------o \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n \8p \11 2 3 4 5 6\8p\, \-\1 \+\8----k-----------------------l \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=1 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p \10 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p \1-1 0 1\ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \1-1 -1 0 1\ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \1-1 -1 -1 0 1\ \ \ \ \8p\, \-\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p \10 -1 -1 -1 0 1\8p\, \-\1 \+\8 \+\1 Whereas the columns of the \ Table 5.8 are irregular and ele-\A \- \+ \+ ments \ of each \ column must \ be found \ separately, columns of the \- \+ \+ Table 5.9 \ repeat, they are only \ shifted in each column \ one row \- \+ \+ down, similarly as the elements \ of their parent matrix. They can \- \+ \+ be easily found \ by multiplying the matrix of \ Euler functions by \-\/ \+ \+ the matrix \4T \1from the left.\, \- \+ \+ \45.9 Partition orbits in m dimensional cubes\, \-\1 \+ \+ Restricted partitions have \ a geometric interpretation: They \-\/ \+ \+ are orbits of n dimensional \ plane complices truncated into cubes \-\/ \+ \+ with the sides (m -1) as on Fig. 5.3.\, \- \+ \+ We can count orbits even in cubes. It \ is a tedious task, if \, \- \+ \+ there are not applied some special techniques, since their number \, \- \+ \+ depends on the size of the cube. For \ example for the 3 dimensio-\A \- \+ \+ nal space we get orbits as in Table 5.10.\, \- \+ \+ \4Table 5.10 Orbits in 3 dimensional cubes\, \-\1 \+\8-----------i------i------i-----------i-------------o \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1Edge size \8p \10 \8p \11 \8p \12 \8p \13\ \ \ \ \ \ \ \ \ \ \ \8p\, \-\1 \+\8-----------k------k------k-----------k-------------l \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=0 \8p \1000 \8p \1000 \8p \1000 \8p \1000\ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 1 \8p p \1100 \8p \1100 \8p \1100\ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p p \1110 \8p \1200,110 \8p \1210,110\ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p p \1111 \8p \1210, 111 \8p \1300,210,111\ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p p p \1220, 211 \8p \1310,220,211\ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p p p \1221 \8p \1320,311,221\ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p p p \1222 \8p \1330,321,222\ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 7 \8p p p p \1331,322\ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 8 \8p p p p \1332\ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p \1 9 \8p p p p \1333\ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p \+\1 \+\2 \1 The equation 5.2 can be applied to cubes. It shows their im-\A \-\8 \/ \+\1 \+\2 \1portant property, \ they are symmetrical along \ the main diagonal, \-\8 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \1going from \ the center of \ the coordinates , the \ simplex n till \-\8 \+\1 \+ the most distant \ vertex of the cube which \ all n coordinates are \-\8 \+\1 \+ (m-1). The diagonal \ of the cube is represented \ on Table 5.10 by \-\8\/ \+\1 \+ k indices. Moreover, a cube is convex, therefore if \, \-\8 \+\1 \+ \ M \9< \1mn/2 p(m,n,M) \9> \1p(m,n,M-1) (5.9)\, \-\8 \+\1 \+ and if M \9> \1mn/2 p(m,n,M) \9< \1p(m,n,M-1) (5.9a)\, \- \+ \+ Here we \ see the importance \ of restricted partitions. \ From \-\/ \+ \+ Table 5.10, \ we can find \ the reccurence, which \ is given by \ the \- \+ \+ fact that in a greater cube the \ lesser cube is always present as \- \+ \+ its base. \ To it new orbits \ are added which are \ on its enlarged \- \+ \+ sides. But it is inough to \ know orbits of one enlarged side, be-\A \- \+ \+ cause the \ other sides are \ formed by these \ orbits. The enlarged \- \+ \+ side \ of a n dimensional \ cube is \ (n - 1) \ dimensional cube. The \-\/ \+ \+ reccurence relation for partitions in cubes is thus\, \- \+ \+ \ \ \ \ \ \ \ \ \ \ p(m,n,M) = p(m-1,n,M) + p(m,n-1,M) (5.10)\, \- \+ \+ This reccurence will be explained later more thoroughly.\, \- \+ \+ \45.10 Generating functions of partitions\ in cubes\, \-\1 \+ \+ The generating function of \ partitions is simply the genera-\A \-\/ \+ \+ ting function of infinite cube \ in the Hilbert space, which sides \-\/ \+ \+ have different meshes:\, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ 2\ \ \ \ \ \ \ \ \ \ \98 \1parts 1: (1 + t + t .. t )\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ 1\ \ \ \ \ 1 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ 2\ \ \ \ \ \98 \1parts 2: (1 + t + t .. t )\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ 2\ \ \ \ \ 2 \+\1 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \1parts \98\1: (1 + t )\, \-\9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8 \+\1 \+ When the multiplications for all parts are made and terms on \-\/ \+ \+ consecutive plane simplices counted, we get:\, \- \+ \+\2\ \ \ \ \ 1\ \ \ \ \ 1\ \ \ \ 2\ \ \ \ \ 1 \11 + t + [t + t ] +[t ...\, \-\2\ \ \ \ 1\ \ \ \ \ 2\ \ \ \ 1\ \ \ \ \ 3 \+\1 \+ For restricted partitions, the generating \ function is modi-\A \-\/ \+ \+ fied \ by cancelling \ unwanted (restricted) \ parts. Sometimes, the \- \+ \+ generating function is formulated in an inverse form. The infini-\A \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1 \1te power series are replaced by \ the differences (1 - t ) . This \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k \+\1 \+ is possible, if we consider t to be only a dummy variable. For e-\A \- \+ \+\9 \1xample, the \ generating function of \ the partitions with \ unequal \-\2\/ \+\1 \+\9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8 \1unrepeated parts is given by the product u(t) = \7P \1(1 = t ).\, \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k=1\ \ \ \ \ \ k \+\1 \+ The mesh \ of the partition \ space is regular, \ it covers all \-\/ \+ \+ numbers. The number of partitions \ is obtained by recursive tech-\A \- \+ \+ niques. But \ it is a very \ complicated function, if \ it should be \-\/ \+ \+ expressed not as one closed formula as the Ramanudjan-Hardy \ fun-\A \-\/ \+ \+ ction is. \ The partitions form a carcass of the space. We will be \-\/ \+ \+ interested, how the mesh of \ partitions is filled into the space, \- \+ \+ which \ all sides \ have unit \ mesh and \ which contains also vector \-\/ \+ \+ strings.\, \- \+ \+ \, \- \= \= \+ \+ \"6. L A T T I C E S O F O R B I T S\, \-\1\ \ \+ \+ \, \- \+\46.1 Partition schemes \+\8 \+\1 \+ Multidimensional plane \ simplices \ are \ complicated objects \-\/ \+ \+ and it is \ necessary to \ find some \ tools how \ to analyze them. To \-\/ \+ \+ draw them is, as we mentioned, impossible, because their parts a-\A \-\/ \+ \+ re layered in our 3 dimensional world over themselves.\, \- \+ \+ We already classified orbits in plane simplices according to \-\/ \+ \+ the number \ k of nonzero parts. This \ shows the dimensionality of \- \+ \+ subsimplices, \ their vertices, \ edges, and \ (k-1) dimensional bo-\A \- \+ \+ dies. \ Then we \ introduced the \ number of \ unit vectors as a tool \- \+ \+ differentiating the simplex. Now we arrange partitions as two di-\A \-\/ \+ \+ mensional tables. These tables we will call partition schemes.\, \- \+ \+ When we analyze a 7 dimensional plane simplex with m = 7, we\, \- \+ \+ can start with its 3 dimensional subsimplices (Fig. 6.1). We see\, \- \+ \+ that they \ contain \ points \ \ corresponding \ to \ partitions: 7,0,0; \-\/ \+ \+ 6,1,0; 5,2,0; 4,3,0; 5,1,1; 4,2,1; 3,3,1; 3,2,2. These partitions\, \- \+ \+ are \ connected with \ other points \ of the \ simplex by circles. In \-\/ \+ \+ higher dimensions the circles become spheres and that is the rea-\A \-\/ \+ \+ son, why we call a partition an orbit.\, \- \+ \+ Arranging partitions into \ tables, the column classification \-\/ \+ \+ is made according \ to the number of \ nonzero parts of partitions. \-\/ \+ \+ We need \ another classifying symptom \ for rows. This \ will be the \-\2 \+\1 \+ length of the \ longest vector m . From all \ partition vectors ha-\A \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \+\1 \+ ving the same dimensionality, \ the longest vector is the vector \-\/ \+ \+ with the longest \ first vector. It dominates them. \ But there can \- \+ \+ exist \ longer orbits \ nearer to \ the surface \ of the simplex. For \- \+ \+ example, vector \ (4,1,1) has equal \ length as (3,3,0) \ and vector \- \+ \+ (4,1,1,1,1) is \ shorter than (3,3,2,0,0). Such \ an arrangement is \- \+ \+ on \ Table 6.1. \ We see \ \ orbits with \ 3 nonzero parts \ inside the \-\/ \+ \+ 3 dimensional simplex, with 2 nonzero parts \ on its sides. Orbits \-\/ \+ \+ with 4 nonzero parts are inside \ tetrahedrons, it is on a surface \- \+ \+ in the fourth dimension. \ There are partitions: 4,1,1,1; 3,2,1,1; \- \+ \+ 2,2,2,1. Similarly, \ we can fill \ columns corresponding to \ higher \-\/ \+ \+ dimensions.\, \- \+ \+ The rows of partition \ schemes classify partitions according \-\2\/ \+\1 \+ to the length of the first longest vector \4e \1. It can be shown \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \+\1 \+ easily that all vectors in higher rows are longer than vectors of\, \- \+ \+ lover rows in corresponding columns. In \ the worst case, it is gi-\A \-\/ \+ \+ ven by the difference \, \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ 2 \1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (x + 1)\ + (x - 1)\ > 2x\, \- \+ \+ We \ can \ consider \ that \ a \ 3 dimensional \ plane \ simplex is \-\/ \+ \+ a truncated 7 dimensional simplex and complete the columns of the \- \+ \+ Tab. 6.1 \ by the corresponding partitions. \ We get a crossec-\A \-\/ \+ \+ tion through the \ 7 dimensional plane. The \ analysis is not \ per-\A \-\/ \+ \+ fect, an \ element is still \ formed by 2 orbits, \ but nevertheless \- \+ \+ the scheme \ gives an insight into \ such a high dimensional space. \-\/ \+ \+ We will study therefore partitions schemes thoroughly.\, \- \+ \+ The number of nonzero vectors in partitions will be given as \, \- \+ \+ n, the size of the first vector as \ m. Zeroes will be not written \-\/ \+ \+ to spare work. The bracket (m,n) means all partitions of the num-\A \-\/ \+ \+ ber m into \ at \ most \ n parts. \ Because \ we \ write a partition as \-\/ \+ \+ a vector, we allow zero parts in a partition. \, \- \+ \+ \4\ \ Table 6.1 Partition scheme (7,7)\, \-\1 \+ \+\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ n\ \8p \11 2 3 4 5 6 7 \8p \7S\, \-\2\ \ \ k \+\8\ \ ------k-----------------------------k--- \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ m\ 7 \8p \11 \8p \11 \, \-\2\ \ \ k\ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 6 \8p \11 \8p \11\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 5 \8p \11 1 \8p \12\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 4 \8p \11 1 1 \8p \13\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 3 \8p \12 1 1 \8p \14\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 2 \8p \11 1 1 \8p \13\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 1 \8p \11 \8p \11\, \- \+\8\ \ ------k-----------------------------k--- \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \7\ \ S \8p \11 3 4 3 2 1 1 \8p \111\, \- \+ \+ \4\ 6.2 Construction of partition schemes\, \-\1 \+ \+ A partition scheme can be divided into four blocks. Diagonal \-\/ \+ \+ blocks repeat the Table 6.1, \ once written in the transposed form \- \+ \+ for n > m/2. Odd \ and even schemes behave differently, \ as it can \-\/ \+ \+ be seen on Tables 6.2 and 6.3\, \- \+ \+ \46.2 Partition scheme m = 14 \, \-\1 \+\8-----i-----------------------i---------------------------------o \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n \ 1 2 3 4 5 6 \ 7 8 9 10 11 12 13 14 \, \-\8-----k-----------------------k---------------------------------l \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=14 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 13 \8p \11 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 12 \8p \11 1\ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 11 \8p \11 1 1\ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 10 \8p \11 2 1 1\ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 9 \8p \11 2 2 1 1\ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 8 \8p \11 3 3 2 1 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 7 \8p \11 3 4 3 2 \8p \11 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\1 \+\8-----k-------i---------------k---------------------------------l \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p p \1* * * * \8p \12 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \ \12 * * * \8p \13 2 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ m---o\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \ \13 * * \8p \14 3 2 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ m---o\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p p \12 * \8p \13 3 2 2 1 1\ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m-------l\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ `\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p ` \11 1 1 1 1 1 1\ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ `\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ `\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ `\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 1 \8p ` \11\ \ \ \8p\, \-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ` \+-----k---------------------------\ -----------------------------l \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ `\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \7 S \8p \11 7 16 23 23 20 \8`\115 \ 11 7 5 3 2 1 1\, \- \+ \+ In the left lower block nonzero elements * can be placed on-\A \-\/ \+ \+ ly over the line which gives a sufficient great product mn to place \- \+ \+ all units \ into the corresponding \ Ferrers graphs and \ their sums \- \+ \+ must agree not \ only with row and column \ sums, but with diagonal \- \+ \+ sums, as \ we show below. This \ can be exploited for \ the calcula-\A \-\/ \+ \+ tions, together with the rules for restricted partitions. \, \- \+ \+ \46.3 Partition scheme m = 15\, \-\1 \+\8-----i---------------------------i------------------------------o \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n \8p \11 2 3 4 5 6 7 \8p\18 9 10 11 12 13 14 15\ \ \8p\, \-\1 \+\8\ \ \ \ \ j---------------------------k------------------------------l \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=15 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 14 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 13 \8p \11 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 12 \8p \11 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 11 \8p \11 2 1 1\ \ \ \ \ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 10 \8p \11 2 2 1 1\ \ \ \ \ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 9 \8p \11 3 3 2 1 1\ \8p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 8 \ 1 3 4 3 2 1 \ 1\, \-\8-----k-------i-------------------k------------------------------l \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 7 \8p p \1* * * * * \8p \11 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p p \1* * * * * \8p \12 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \ \1* * * * * \8p \13 2 1 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ m---o\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p p \1* * * * \8p \14 3 2 1 1\ \ \ \ \ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ m--o\ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p p \1* * * \8p \13 3 2 2 1 1\ \ \ \ \ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ m------------l\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p p \11 1 1 1 1 1 1\ \ \ \ \8p\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 1 \8p p \11\8p\, \-\1 \+\8-----k---------------------------k------------------------------l \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p p \11 7 19 27 30 26 21 \8p\115 11 7 5 3 2 1 1\8p\, \-\ \ \ \ \ p \+\1 \+ \4\ 6.3 Lattices of orbits\, \-\1 \+ \+ Partition orbits are spheres which radius r is determined by \-\/ \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ 1/2 \1the Euclidean length of \ the corresponding vector: r = (\7S\1p ) . \-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j \+\1 \+ Radiuses of some partition \ orbits coincide, for example r(3,3,0) \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1/2 \1= r(4,1,1) = (18) . It is \ thus impossible to determine distan-\A \- \+ \+\2 \1ces betweem \ orbits using these \ radii, the distance \ between two \-\/ \+ \+\2 \1different orbits can not be zero.\, \- \+ \+ We have shown in Chapter 5.4 \ that one orbit can be obtained \-\/ \+ \+ from another by shifting just two \ vectors, one up and other down \- \+ \+ on the number scale. We can imagine that both vectors collide and \- \+ \+ exchange their values as two \ particles of the ideal gas exchange \- \+ \+ their \ energy. If \ we limit \ the result \ of such \ an exchange \ on \- \+ \+ 1 unit, we can consider such two \ orbits to be the nearest neigh-\A \- \+ \+ bor orbits. We connect them in \ the scheme by a line. Some orbits \- \+ \+ are thus connected with many \ neighbour orbits, other have just one \- \+ \+ neighbour as on \ Fig.6.1. Orbits (3,3,0) and (4,1,1) \ are not nea-\A \-\/ \+ \+ rest neighbours, because they must be transformed in two steps:\, \- \+ \+ (3,3,0) <---> ((3,2,1) <---> (4,1,1) or (3,3,0) <---> (4,2,0) <-- \- \+ \+ -> (4,1,1). \- \+ \+ Partition schemes \ are generally not \ suitable for construc-\A \-\/ \+ \+ tion of \ orbit lattices, because at \ m=n > 7 there appear several \- \+ \+ orbits on some \ table places. It is necessary \ to construct 3 di-\A \-\/ \+ \+ mensional lattices to show all existing connections.\, \- \+ \+\2 \1 Sometimes, there are given \ stronger conditions on processes \-\2\/ \+\1 \+\2 \1going at exchanges, \ namely that at each collision \ the number of \-\2 \+\1 \+\2 \1empty parts must change, as if they were information files which can \-\2 \+\1 \+\2 \1be only \ joined into one \ file or separated \ into more files, \ or \-\2 \+\1 \+\2 \1a part of \ a file transferred into \ an empty file. \ Also here the \-\2 \+\1 \+\2 \1nearest \ neighbour is \ limited on \ unifying of \ just of 2 files or \-\2 \+\1 \+\2 \1splitting a file into two. Here \ the path between two orbits must \-\2 \+\1 \+ be longer, for example: (3,3,0) <---> (6,0,0) <---> (5,1,0) <---> \-\/ \+ \+ (4,1,1) or (3,3,0) <---> (3,2,1) <---> (5,1,0) <---> (4,1,1).\, \- \+ \+ Such lattices were demanded for information files.\, \- \+ \+ In the lattice it is possible to count the number of nearest \-\/ \+ \+ neighbours. If \ we investigate the number \ of one unit neighbours \- \+ \+ or connecting lines between columns \ of partition schemes, we ob-\A \-\/ \+ \+ tain an interesting Table 6.4\, \- \+ \+ \4Table 6.4 Right hand one unit neighbours of partition orbits\, \-\1 \+ \+\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n \8p \11 2 3 4 5 6 \8p \7S\, \-\1 \+\8----k-----------------------k-- \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1m=2 \8p \11 \8p \11\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p \11 1 \8p \12\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \11 2 1 \8p \14\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \11 3 2 1 \8p \17\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p \11 4 4 2 1 \8p\112\, \-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 7 \8p \11 5 6 4 2 1 \8p\119\, \- \+\8----,-----------------------k-- \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \7D\1(7-6)0 1 2 2 1 1 \8p \17\, \- \+ \+ The number of neighbours is the \ sum of the number of parti-\A \-\/ \+ \+\2\ \ \ \ \ \ \ \ m \1tions \7S \1p(k-2)\2. \1The row differences of \ the Table 6.4 are num-\A \-\2\ \ \ \ \ \ \ k=2\/ \+\1 \+\2 \1bers of \ restricted partitions. It is \ explained easily. Each one \-\2 \+\1 \+\2 \1unit right \ neighbour shifts just \ one pair of \ vectors together. \-\2 \+\1 \+\2 \1When we add 1 to all partitions \ p(m-1) and find all their neigh-\A \-\2 \+\1 \+\2 \1bours similar to their neighbours, there will remain unperturbed \-\2 \+\1 \+\2 \1pairs of vectors which number is determined by the number of res-\A \-\2 \+\1 \+\2 \1tricted partitions \ p(m-2) which count \ changes of zero \ vectors. \-\2 \+\1 \+\2 \1Thus the number of all right neighbours in the lattice is the sum \-\2 \+\1 \+\2 \1of the number \ of partitions. To find all \ neighbours, we must add \-\2 \+\1 \+\2 \1neighbours inside \ columns. The number \ of elements in \ columns is \-\2 \+\1 \+\2 \1the number of partitions into \ exactly n parts p(m,n), the diffe-\A \-\2 \+\1 \+\2 \1rence in each column must \ be decreased by 1, but, unfortunately, \-\2\/ \+\1 \+\2 \1there are more connections, for example: 611\8--\1521\8--\1431\8--\1332\, \-\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ p\ \ \ \ p \+\1 \8 m--\ \1422\8---. \ \ \, \-\1 \+\8 \+\1 These connections must be counted separately.\, \- \+ \+ \46.4 Diagonal differences on lattices\, \-\1 \+ \+ On lattices, we can count orbits on diagonals, consecutively \- \+ \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k \1from the \ main diagonal, which \ counts orbits [n \ - k] 1ö. \ Their \- \+ \+\2 \1Ferrers graphs have a form *\, \- \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ * \+ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *\, \- \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ * * * * * \+ \, \- \+ \+ Other diagonals counts partitions which have in this layer lesser \- \+ \+ numbers of units, the other are inside this base.\, \- \+ \+ The corresponding table is as follows\, \- \+ \+ \, \- \+ \+ \4Table 6.5 Diagonal differences of partitions\, \-\1 \+ \+ \8-----i-------------------------------------i----------------------\, \-\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1k \8p \11 2 3 4 5 6 7 8 9\ \ \8p\ \ \7S\, \-\1 \+\8-----k-------------------------------------k----------------------- \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1n= 1 \8p \11\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \11\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 2 \8p \12\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \12\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 3 \8p \13\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \13\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 4 \8p \14 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \15\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 5 \8p \15 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \ \17\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 6 \8p \16 3 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \111\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 7 \8p \17 4 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \115\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 8 \8p \18 5 6 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \122\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 9 \8p \19 6 8 6 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \130\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 10 \8p \110 7 10 9 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \142\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 11 \8p \111 8 12 12 11 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\ \156\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 12 \8p \112 9 14 15 16 9 2\ \ \ \ \ \ \ \ \ \ \8p\ \177\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 13 \8p \113 10 16 18 21 16 7\ \ \ \ \ \ \ \ \ \ \8p\1101\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 14 \8p \114 11 18 21 26 23 18 4\ \ \ \ \ \ \8p\1135\, \-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1 15 \8p \115 12 20 24 31 30 29 12 3\ \ \8p\1176\, \- \+ \+ The initial k column values are corresponding to:\, \- \+ \+ 1. 1n\, \- \+ \+ 2. 1(n-3)\, \- \+ \+ 3. 2(n-5)\, \- \+ \+ 4. 3(n-7)\, \- \+ \+ 5. 5(n-9) + 1\, \- \+ \+ 6. 7(n-11) + 2\, \- \+ \+ The values \ in brackets, till \ k = 6, are \ the numbers free \-\/ \+ \+ for partitions inside the L frame \ having (2k-1) units. At higher \- \+ \+ diagonal layers, their partitions start in the third column. Par-\A \-\/ \+ \+ titions 4, 4, 4 and \ 3, 3, 3, 3, for n=12, are \ counted in the 7. \-\/ \+ \+ layer. At n=13, this layer counts seven partitions:\, \- \+ \+ 5, 5, 3\, \- \+ \+ 5, 4, 4 \, \- \+ \+ 4, 4, 4, 1\, \- \+ \+ 4, 4, 3, 2\, \- \+ \+ 4, 3, 3, 3\, \- \+ \+ 3, 3, 3, 3, 1\, \- \+ \+ 3, 3, 3, 2, 1\, \- \+ \+ The multipliers of the first columns are \ the number of par-\A \-\/ \+ \+ titions p(k-1). Permutations of free \ units of the frame multiply \- \+ \+ the partition inside it. But \ this is somewhat complicated as for \-\/ \+ \+ 21 partitions of 13 in the 5. layer:\, \- \+ \+ 8, 5 6, 5, 1, 1\, \- \+ \+ 7, 4, 2 5, 4, 2, 1, 1\, \- \+ \+ 7, 3, 3 5, 3, 3, 1, 1\, \- \+ \+ 6, 3, 2, 2 4, 3, 2, 2, 1, 1\, \- \+ \+ 5, 2, 2, 2 3, 2, 2 ,2, 2, 1, 1\, \- \+ \+ \, \- \+ \+ 7, 5, 1 5, 5, 1, 1, 1\, \- \+ \+ 6, 4, 2, 1 4, 4, 2, 1, 1, 1\, \- \+ \+ 6, 3, 3, 1 4, 3, 3, 1, 1, 1\, \- \+ \+ 5, 3, 2, 2, 1 3, 3, 2, 2, 2, 1, 1, 1\, \- \+ \+ 4, 2, 2, 2, 2, 1 2, 2, 2, 2, 2, 2, 1, 1, 1\, \- \+ \+ The basic frames are: 8, 1; 7, 1, 1; 6, 1, 1, 1 and 5, 1, 1, \-\/ \+ \+ 1, 1. Aside, \ there is the conjugate partition 3, \ 3, 3, 1, 1, 1, \- \+ \+ 1 to the partition \ 7, 3, 3, which was not \ counted by these fra-\A \- \+ \+ mes. There appears the basic \ partition of this diagonal layer 3, \-\/ \+ \+ 3, 3.\, \- \+ \+ \46.5 Inversed schemes\, \-\1 \+ \+ The schemes \ have form of \ the matrix in \ the lower diagonal \- \+ \+ form \ with unit \ diagonal. Therefore, \ they have \ inverses. It is \- \+ \+ easy to find them, e. g. for n=7\, \- \+ \+ \4\ Table 6.5 Partition scheme (7,7)\ and its inversion\, \-\1 \+ \+\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ n\ \8p \11 2 3 4 5 6 7 \8p \, \-\2\ \ \ k \+\8\ \ ------k---------------------------k--- \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ m\ 7 \8p \11 \8p \11 \, \-\2\ \ \ k\ \ \ \ \8p \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 6 \8p \11 \8p \10 1\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 5 \8p \11 1 \8p \10 -1 1\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 4 \8p \11 1 1 \8p \12 -1 -1 1\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 3 \8p \12 1 1 \8p \1-2 2 0 -1 1\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \1\ \ 2 \8p \11 1 1 \8p \10 0 0 0 0 1\, \-\8\ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p \+\1 \+ The partitions in \ rows must be balanced with \ other ones by \-\/ \+\8 \+\1 elements of inverse columns. The third column includes and exclu-\A \- \+\8 \+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4\ \ \ 3\ \ \ \ \ \ \ 2\ 3\ \ \ \ \ \ \ \ \ \ \ 5 \1des 331 and 322 with 3211 and 31 ; 2 1 and 2 1 with 2* 21 , res-\A \-\/ \+\8 \+\1 pectively.\, \- \+ \+ \46.6 Generalized lattices\, \-\1 \+ \+ We will use the notion \ of lattices also for possible trans-\A \-\/ \+ \+ formations of points having \ specific properties between themsel-\A \- \+ \+ ves, for example between all \ 10 permutations of a 5 tuple compo-\A \- \+ \+ sed \ from 3 symbols \ of one \ kind and \ 2 symbols of another kind. \- \+ \+ When the neighbours must differ \ only by one exchange of the po-\A \- \+ \+ sition of only one pair of symbols of both kind we obtain lattice \- \+ \+ as on Fig. 6.4. The lattice \ forms a simple triangle. But for the \- \+ \+ simultaneous exchange of two pairs the 10 points are arranged in-\A \-\/ \+ \+ to aöcomplicated pattern as on Fig.6.5, known as Pettersen graph.\, \- \+ \+ Lattices are formed by vertices of n dimensional cubes. The-\A \-\/ \+ \+ re nearest \ vertices differ by only \ one coordinate. The lattices \- \+ \+ of 2 and \ 3 dimensional cubes are \ on Fig. 6.5. \ Compare lines of \-\/ \+ \+ the graphs with a real cube and try to draw 4 dimensional cube.\, \- \+ \+ A \ classical \ example \ on \ relation \ lattices is Aristoteles \- \+ \+ attribution of four properties: warm, cold, dry and humid to four \- \+ \+ elements: \ fire, air, \ water and \ earth, respectively. \ It can be \- \+ \+ arranged in a form \, \- \+ \+ air\, \-\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u------i-------o \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ p \+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ p \1 \ warm\ humid\, \-\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j------k-------l \+\1\ \ \ \ \ \ \ \ \ \ \ \ \ fire\ \ \8p\ \ \ \ \ \ p\ \ \ \ \ \ \ p\ \ \1water \+\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ p \1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ dry \8p\ \1cold\, \-\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m--------------. \+\1 \+ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ earth\, \- \+ \+ The opposite elements have always only two nearest properti-\A \-\/ \+ \+ es. The diagonal properties exclude themselves. Something can not \- \+ \+ be \ simultaneously \ warm \ and \ cold, \ \ or \ humid \ and \ dry. \ More \- \+ \+ precisely, \ it is \ necessary to \ draw a borderline \ between these \-\/ \+ \+ properties. Even \ water vapor can be \ dry as well as \ wet. It de-\A \-\/ \+ \+ pends on its saturation.\, \- \+ \+ Try to draw the lattice of relations and compare it with the \-\/ \+ \+ lattice of \ 4 dimensional cube. The elements \ correspond to layer \- \+ \+ 1, the properties to layer 3. The layer 2 is truncated from 6 to \-\/ \+ \+ 4 combinations.\, \- \+ \+ \, \- \+ \+ \- \+ \+\2 \1\, \-\2 \+\1 \+ \- \+ \+ \, \-\2 \+\1 \+\2 \1\, \- \+ \+ \, \- \+ \+ \- \+ \+ \, \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \, \- \+ \+ \- \+\4 \+\1 \-\8 \+\7 \+\8 \1 \-\2 \+\8 \+\1 \-\8 \+\1 \+ \-\8 \+\1 \+ \, \-\8 \+\1 \+\2 \1\, \-\2 \+\8 \+\2 \1\, \-\2 \+\8 \+\1 \-\8\/ \+\1 \+ \-\8 \+\1 \+ \, \-\/ \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \, \- \+ \+ \-\/ \+ \+ \- \+ \+ \- \+ \+ \- \+ \+\2 \1 \-\2\/ \+\1 \+\2 \1\, \-\9 \+\1 \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \7\, \-\1 \+\8 \+\1 \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \- \+ \+ \-\/ \+ \+ \-\2 \+\1 \+\2 \1 \-\2\/ \+\1 \+\2 \1 \-\2 \+\1 \+\2 \1 \-\2\/ \+\1 \+\2 \1 \-\2\/ \+\1 \+ \- \+ \+ \-\/ \+ \+ \, \- \+ \+\2 \1 \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+ \+ \-\/ \+ \+ \- \+ \+ \, \-\/ \+ \+ \-\2\/ \+\1 \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+\2 \1 \-\2 \+\1 \+\2 \1 \-\2 \+\1 \+\2 \1 \-\2 \+\1 \+\2 \1 \-\2 \+\1 \+\2 \1 \-\2 \+\1 \+\2 \1 \- \+ \+\2 \1 \-\/ \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \4\, \-\1 \+ \+ \8\, \-\1 \+ \+\2 \1\, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \- \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \-\8 \+\1 \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \-\2 \+\1 \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \- \+ \+ \, \- \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \-\/ \+ \+ \- \+ \+ \-\/ \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \- \+\8 \+\1 \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \, \-\8 \+\1 \+ \4\, \-\1 \+ \+ \-\/ \+ \+ \- \+ \+ \-\/ \+ \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \-\2 \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \-\2 \+\1 \+ \-\2 \+\1 \+ \-\2 \+\1 \+ \-\2 \+\1 \+ \, \-\2 \+\1 \+ \4\, \-\1 \+ \+ \- \+ \+ \-\2 \+\1 \+ \-\2 \+\1 \+ \, \-\2 \+\1 \+ \4\, \-\1 \+\8 \+\1 \, \- \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+\2 \+\1 \, \-\2 \+\1 \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+\2 \1 \-\2 \+\1 \+ \- \+ \+\2 \1 \-\2 \+\1 \+ \- \+ \+ \, \- \+ \+\2 \1 \- \+ \+ \- \+ \+\2 \1 \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \- \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \- \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \- \+ \+ \- \+ \+ \, \- \+ \+ \4\, \-\1 \+\8 \+\1 \, \- \+\8 \+\1 \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+ \+ \, \- \+\8 \+\1 \, \- \=