Original design by Curtis Greene, Eugenie Hunsicker, 1990
Updates by John Dollhopf, Sam Hsiao, Curtis Greene, 1992-94
Partially supported by grants from the National Science Foundation.
This document describes a package designed to generate, display, and explore partially ordered sets (also known as "posets"), with an emphasis on topics of current interest in combinatorics. The package has two distinctive features: (1) a large repertoire of standard examples (Boolean lattices, subword posets, lattices of partitions, distributive lattices, Young's lattice, Bruhat orders, etc.), and (2) the ability to generate a poset directly from its formal definition, i.e., from a Mathematica program giving its "covering function". The latter approach permits a compact definition of posets whose structure is extremely complex, or unknown. Posets of 100-200 elements can typically be generated and displayed within a couple of seconds (on current Pentiums, for example).
A poset may also be defined by listing its covering pairs, or any generating set of ordered pairs, or by giving the incidence matrix for such a relation. Also included are some standard constructions of new posets from old: product and sum, subposet, distributive lattice of order ideals, etc.
Once a poset has been generated, a large number of tools are available for investigating its combinatorial structure. In the present version of the package the selection of tools implemented reflects the authors' interests: enumerative invariants, chains, multichains, antichains, rank generating functions, Mobius functions, linear extensions, and related topics. A good reference for the underlying combinatorics is Richard Stanley's book, Enumerative Combinatorics (Wadsworth, 1986).
The package is available in both notebook and plain ascii text formats:
Eventually we hope to have full documentation available on these pages. For now we can offer only the following:
The authors welcome comments, questions, bug reports, criticisms, and other suggestions.
Such communications should be directed to cgreene@haverford.edu,
or to Curtis Greene, Department of Mathematics, Haverford College, Haverford PA 19041.
Still a bit under construction. 