# POSETS

POSETS: A Mathematica Package for Studying Posets. 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 quickly. 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 current package is version 3.00 (posets300.m), updated in summer 2012, designed to be compatible with Mathematica V8.0.

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