A general algorithm for the MacMahon Omega operator
type: Item
instance of: scholarly article
title: A general algorithm for the MacMahon Omega operator
zbMATH Open document ID: 1036.05005
author: Guo-Niu Han
published in: Annals of Combinatorics
publication date: 2004-03-11T00:00:00Z
review text: Let \(F\) be formally defined by \[ F=\sum_{s_1=-\infty}^{\infty} \cdots \sum_{s_r=-\infty}^{\infty} A_{s_1,\dots,s_r} \lambda_1^{s_1}\cdots \lambda_r^{s_r}. \] In his classic ``Combinatory analysis'' MacMahon introduced the operator \(\Omega\) acting on \(F\) as \[ \Omega(F)=\sum_{s_1=0}^{\infty} \cdots \sum_{s_r=0}^{\infty} A_{s_1,\dots,s_r}. \] The usefulness of the operator \(\Omega\) lies in its power to solve problems dealing with Diophantine inequalities. For example, consider the problem of having to compute the generating function \[ \sideset\and{'}\to \sum_{t_1,t_2,\dots,t_n\geq 0} q^{t_1+\cdots+t_n} \] where the prime over the sum indicates that the \(t_i\) must satisfy \(m\) Diophantine inequalities \[ a_{i1}t_1+ \cdots+a_{in}t_n\geq 0,\qquad i=1,\dots,m. \] The \(\Omega\) operator easily solves this problem since \[ \begin{aligned} \sideset\and{'}\to \sum_{t_1,t_2,\dots,t_n\geq 0} q^{t_1+\cdots+t_n} &= \Omega \sum_{t_1,t_2,\dots,t_n\geq 0} q^{t_1+\cdots+t_n}\prod_{i=1}^m \lambda_i^{\sum_j a_{ij}t_j} \\ &=\Omega \prod_{j=1}^n \frac{1} {1-q\lambda_1^{a_{1j}}\cdots\lambda_m^{a_{mj}}}.\end{aligned} \] Now the real problem lies with the actual implementation of the operator \(\Omega\). \textit{G. E. Andrews}, \textit{P. Paule} and \textit{A. Riese} [MacMahon's partition analysis. VI: A new reduction algorithm, Ann. Comb. 5, 251--270 (2001; Zbl 1027.05005)] recently developed an algorithm for computing \[ \Omega \frac{\lambda^a} {(1-x_1 \lambda^{j_1}) \cdots(1-x_n\lambda^{j_n})(1-y_1\lambda^{-k_1}) \cdots(1-y_m\lambda^{-k_m})} \] and implemented this in their Mathematica package \textit{Omega}. In the paper under review, Han considers the more general problem of evaluating \[ \Omega \frac{U (\lambda)}{A(\lambda)B(1/\lambda)} \] where \(U(t)\) is a Laurent polynomial and \(A(t)\) and \(B(t)\) are arbitrary polynomials. The author solves the above problem without having to determine first the roots of the polynomials \(A(t)\) and \(B(t)\). This allows his Maple…
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 2055154
zbMATH Keywords: partition analysis
zbMATH Keywords: Diophantine inequalities
reviewed by: S. Ole Warnaar
cites work: GenOmega
describes a project that uses: GenOmega
describes a project that uses: Maple
describes a project that uses: Mathematica
describes a project that uses: Omega
MaRDI profile type: Publication
full work available at URL: s00026-003-0197-8
OpenAlex ID: W2059684321
DOI: 10.1007/S00026-003-0197-8
description: scientific article; zbMATH DE number 2055154