Omega - Partition Analysis

Short Description

Omega is a Mathematica implementation of MacMahon's Partition Analysis carried out by Axel Riese, a Postdoc of the RISC Combinatorics group. It has been developed together with George E. Andrews and Peter Paule within the frame of a project initiated by Andrews on the occasion of his sabbatical at RISC in spring 1998.

Partition Analysis is a computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations, respectively. But as a matter of fact, MacMahon's ideas have not received due attention with the exception of work by Richard Stanley. The object of the Omega project is to change this situation by demonstrating the power of MacMahon's method in current combinatorial research.

Registration and Legal Notices

The source code for this package is password protected. To get the password send an email to Peter Paule. It will be given for free to all researchers and non-commercial users.

A Note on Encoded Files

This package contains one or more Mathematica input files which are encoded. Those files cannot be read or modified directly as plain text, but can be loaded into Mathematica just like any normal input file (i.e., with <<"file" or Get["file"]). There is no need (and also no way) to decode them by using additional software or a special key.

If loading an encoded file causes a syntax error, open it with a text editor and remove any blank lines at the beginning (for some reason your Mac could have inserted them silently...).

The Package

The Omega package consists of the Mathematica input file

Omega2.m (encoded)

and the documentation files

Readme.txt (ASCII)
OmegaDemo.nb (Mathematica notebook)

You can also download the notebook that I presented at the Special Functions 2000 conference in Tempe, Arizona:

OmegaTempe.nb (Mathematica notebook)

Screenshot

Click here for MacMahon's "Solid Partitions on a Cube" example.

Literature

Partition Analysis has been originally described in

P.A. MacMahon, Memoir on the theory of the partition of numbers - Part I, Phil. Trans. 187 (1897), 619-673,
P.A. MacMahon, Memoir on the theory of the partition of numbers - Part II, Phil. Trans. 192 (1899), 351-401,
P.A. MacMahon, Memoir on the theory of the partition of numbers - Part III, Phil. Trans. 205 (1906), 35-58,

and has been recapped in

P.A. MacMahon, Collected Papers, Vol. 2, Number Theory, Invariants, and Applications (G.E. Andrews, ed.), MIT Press, Cambridge, 1986.

How to refer to the Omega package?

The first description of the package can be found in the article

G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis III: The Omega Package, European J. Combin., 22 (2001), 887-904. [pdf]

Further articles of the Omega project:

G.E. Andrews and P. Paule, MacMahon's Partition Analysis IV: Hypergeometric Multisums, Sém. Lothar. Combin., B42i (1999), 1-24. [pdf]
G.E. Andrews, P. Paule, A. Riese, and V. Strehl, MacMahon's Partition Analysis V: Bijections, Recursions, and Magic Squares, in Algebraic Combinatorics and Applications (A. Betten et al., eds.), pp. 1-39, Springer, 2001. [pdf]
G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis VI: A New Reduction Algorithm, Ann. Comb., 5 (2001), 251-270. [pdf]
G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis VII: Constrained Compositions, in q-Series with Applications to Combinatorics, Number Theory, and Physics (B.C. Berndt and K. Ono, eds.), Contemp. Math., Vol. 291, pp. 11-27, Amer. Math. Soc., 2001. [pdf]
G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis VIII: Plane Partition Diamonds, Adv. in Appl. Math., 27 (2001), 231-242. [pdf]
G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc., 64 (2001), 321-329. [pdf]
G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis X: Plane Partitions with Diagonals, SFB Report n. 2004-2, J. Kepler University, Linz, 2004. [pdf]
G.E. Andrews, P. Paule, and A. Riese, MacMahon's Partition Analysis XI: Hexagonal Plane Partitions, SFB Report n. 2004-4, J. Kepler University, Linz, 2004. [pdf]

Versions and Bugs

The current version of the package is 2.48 last updated on Janyary 10, 2008. Please report any bugs and comments to Ralf Hemmecke.