A general algorithm for the MacMahon Omega operator
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 package \textit{GenOmega} to, for example, compute that \[ \Omega\frac{1}{(1+x\lambda+y\lambda^5)(1+a/\lambda)} =\frac{1+ay(1-a)(1+a^2)}{(1+x+y)(1-ax-a^5y)}, \] a result not feasible with \textit{Omega}.
scientific article; zbMATH DE number 2055154