---------------- | Omega Calculus | ---------------- This document briefly describes version 2.x of the Omega package written by Axel Riese. The package mainly provides the functions OR and OEqR which apply the Omega>= and Omega= operator, respectively, w.r.t. a certain variable to an expression. The calling syntax is OR[expr, z] and OEqR[expr, z] or OR[Omega[expr, z]] and OEqR[OmegaEq[expr, z]] where z is a variable (i.e., a Mathematica symbol, an indexed symbol or a subscripted symbol) and expr is of the form LP(z) ----------------------------------- (1 +/- pp_1(z)) ... (1 +/- pp_d(z)) with LP(z) a Laurent polynomial in z, pp_i(z) power products (with integer exponents) in z and some other variables. Several variables can be eliminated in one stroke by calling OR[expr, {z_1, ..., z_n}] and OEqR[expr, {z_1, ..., z_n}] or OR[Omega[expr, {z_1, ..., z_n}]] and OEqR[OmegaEq[expr, {z_1, ..., z_n}]] In this case expr must be of the form LP(z_1,...,z_n) ------------------------------------------------------- (1 +/- pp_1(z_1,...,z_n)) ... (1 +/- pp_d(z_1,...,z_n)) Further information can be found at the Omega homepage ----------------------------------------------------------------------- | http://www.risc.uni-linz.ac.at/research/combinat/risc/software/Omega/ | ----------------------------------------------------------------------- and in the accompanying Mathematica notebook OmegaDemo.nb. Example: ------- For applying Partition Analysis to (a special case of) Lecture Hall partitions one proceeds as follows. In[1]:= <= 4 n2, 2 n2 >= 3 n3, n3 >= 2 n4}, l] Assuming n1 >= 0 Assuming n2 >= 0 Assuming n3 >= 0 Assuming n4 >= 0 1 Omega[-----------------------------------------------, {l , l , l }] 2 1 2 3 a2 l a3 l 3 2 a4 3 (1 - a1 l ) (1 - ------) (1 - ---) (1 - -----) 1 4 2 3 l l l 1 3 2 And now we do the elimination: In[3]:= OR[%] Eliminating l 1 Eliminating l 3 Eliminating l 2 2 3 2 3 2 4 3 6 4 1 + a1 a2 + a1 a2 + a1 a2 a3 + a1 a2 a3 + a1 a2 a3 Out[3]= ------------------------------------------------------------- 4 3 4 3 2 4 3 2 (1 - a1) (1 - a1 a2 ) (1 - a1 a2 a3 ) (1 - a1 a2 a3 a4) ======================================================================= Axel Riese Research Institute for Symbolic Computation J. Kepler University Linz Tel: +43 (0)732 2468 9939 Altenbergerstrasse 69 Fax: +43 (0)732 2468 9930 A-4040 Linz e-Mail: Axel.Riese@risc.uni-linz.ac.at Austria http://www.risc.uni-linz.ac.at/people/ariese/home/ =======================================================================