MacMahon's partition analysis. IX: \(k\)-gon partitions (Q2758195)

This paper deals with the use of partition analysis, a computational technique for dealing with combinatorial problems in connection with systems of linear diophantine equations and inequalities, which is known to become a very powerful method when implemented in computer algebra. Here the authors explain the use of their developed package OMEGA (available at: \url{http://www.risc.uni-linz.ac.at/research/combinat/risc/software/Omega}) leading to a generalization of a classical problem counting the number of triangles with sides of integer length with the generating function NEWLINE\[NEWLINET_3(q):= \sum_{n\geq 3} t_3(n) q^n= \sum^* q^{a_1+ a_2+ a_3},NEWLINE\]NEWLINE where \(t_3(n)\) is the number of non-congruent triangles whose sides are of integer length with perimeter \(n\), and \(\sum^*\) denotes the summation over all positive integers \((a_1,a_2,a_3)\) satisfying \(a_1\leq a_2\leq a_3\) and \(a_1+ a_2> a_3\). Using Omega package, the authors easily obtain NEWLINE\[NEWLINET_3(q)= {q^3\over (1- q^2)(1- q^3)(1- q^4)}.NEWLINE\]NEWLINE \(T_3(q)\) is then generalized to the full generating function \(S_3(x_1,x_2,x_3)= \sum^*= x^{a_1}_1 x^{a_2}_2 x^{a_3}_3\) (with \(\sum^*\) and the \(a_i\)'s as defined above), which is shown to equal NEWLINE\[NEWLINE{x_1x_2x_3\over (1- x_2x_3)(1- x_1x_2x_3)(1- x_1x_2x^2_3)}.NEWLINE\]NEWLINE These results are then generalized to \(T_k(q)\) and \(S_K(x_1,x_2,x_3,\dots, x_K)\) and the main result for \(k\)-gon partitions are as follows:NEWLINENEWLINENEWLINEResults: (a) Let \(k\geq 3\) and \(X_i= x_ix_{i+1}\cdots x_K\), \(1\leq i\leq K\). Then NEWLINE\[NEWLINE\begin{multlined} S_K(x_1,x_2,\dots, x_K)= {X_1\over (1- X_1)(1- X_2)\cdots(1- X_K)}-\\ {X_1 X^{K-2}_K\over 1- X_K}\cdot {1\over (1- X_K)(1- X_{K-2} X_K)(1- X_{K-3} X^2_K)\cdots(1- x_1 X^{K-2}_K)}.\end{multlined}NEWLINE\]NEWLINE (b) For \(K\geq 3\), we have NEWLINE\[NEWLINET_K(q)= {q^K\over (1-q)(1- q^2)\cdots(1- q^k)}- {q^{2K- 2}\over 1- q}\cdot{1\over (1- q^2)(1-q^4)\cdots (1- q^{2k-2})}.NEWLINE\]NEWLINE In \(k\)-gon partitions the constraints on positive integers \((a_1,a_2,\dots, a_K)\) satisfy \(1\leq a_1\leq a_2\leq\cdots\leq a_K\) and \(a_1+ a_2+\cdots+ a_{K-1}> a_K\).NEWLINENEWLINENEWLINEFor part VIII see Zbl 0992.05017.