Implement MacMahon's partition analysis Omega operator

[nthiery]

Consider a multivariate fraction F, mixing parameters and variables
(or possibly just an Eliot fraction, where the denonimators are
binomials). The Omega operator applied on F returns the constant term
of F, under the form of a fraction in the parameters.

A typical application of this tool is to build the generating function
for all the solutions to a system of Diophantine linear equation. It
has also been used in many papers to build closed form formula for
generating series.

Implementations and algorithms:

• [http://www.risc.jku.at/research/combinat/software/Omega/ Mathematica implementation](http://www.risc.jku.at/research/combinat/software/Omega/ Mathematica implementation) by Andrews/Paule/Riese:
  Non-Free. Sources available upon request.

• [http://www.combinatorics.net.cn/homepage/xin/maple/Ell.rar Maple implementation](http://www.combinatorics.net.cn/homepage/xin/maple/Ell.rar Maple implementation)
  by Guoce Xin who realized that Laurent series were the appropriate
  setup for this problem, both conceptually and to derive efficient
  algorithms using explicit partial fraction decomposition, together
  with subtle heuristics for controlling the number of terms ([2];
  see also Zeilberger opinion [3]).

• [http://www-irma.u-strasbg.fr/~guoniu/software/ Maple implementation](http://www-irma.u-strasbg.fr/~guoniu/software/ Maple implementation)
  by Guo-Niu Han, who generalized Xin's algorithm from Eliot
  fractions to any rational fraction [3]

• [http://www.risc.jku.at/research/combinat/software/GenOmega/index.php Mathematica implementation](http://www.risc.jku.at/research/combinat/software/GenOmega/index.php Mathematica implementation) by Wiesinger of Han's algorithm
  The link also point to Wiesinger's master thesis on the topic.

• [http://mupad-combinat.svn.sourceforge.net/viewvc/mupad-combinat/trunk/MuPAD-Combinat/experimental/2006-06-27-Omega.mu MuPAD crude implementation](http://mupad-combinat.svn.sourceforge.net/viewvc/mupad-combinat/trunk/MuPAD-Combinat/experimental/2006-06-27-Omega.mu MuPAD crude implementation) of Xin's algorithm by Thiéry:

  The only reason to mention it here is for the attempts at using
  proper data structures and object orientation; it is my bet that
  those could eventually yield not only much more readable code, but
  also eventually faster. However at this point the heuristics are
  improperly fine tuned, and the code darn slow.

• Links with Schur functions, by Fu and Lascoux [4]

• Sage prototype (Bandlow/Musiker) written at Sage Days 7: ...

[1] http://arxiv.org/abs/math.CO/0408377
[2] http://www.math.rutgers.edu/~zeilberg/Opinion74.html
[3] http://www-irma.u-strasbg.fr/~guoniu/papers/p36omega.pdf
[4] http://arxiv.org/abs/math/0404064

CC: @sagetrac-sage-combinat @jbandlow @sagetrac-gmusiker @zafeirakopoulos

Component: combinatorics

Issue created by migration from https://trac.sagemath.org/ticket/10669

[burcin]

comment:2

I have a Sage implementation of the Omega operator, mainly based on the Andrews/Paule/Riese papers. (I haven't seen the MMA implementation). Maybe Zaf is interested in working on it so it can be included in Sage.

[dkrenn]

comment:3

An implementation can now be found at #22066.

[videlec]

comment:6

closing positively reviewed duplicates