posets : a mathematica packageww3.haverford.edu/math/cgreene/posets/posets300-doc1.pdf · (1) a...

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POSETS : A Mathematica Package PART 1: Basic Commands and Families of Posets Curtis Greene, Eugenie Hunsicker, Department of Mathematics, Haverford College 19041, July 1990 Updates 1992-94: John Dollhopf, Sam Hsiao, Curtis Greene Updates 2008: Erica Greene, Curtis Greene Updates 2010-12: Ian Burnette, Curtis Greene 1. Introduction This document illustrates a Mathematica package designed to generate, display, and explore finite partially ordered sets ("posets"), emphasizing properties of use in combinatorics. The package has two distinctive features: (1) a large repertoire of built-in 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 defining its "covering function". A poset may also be defined by listing its covering pairs, or by giving any generating set of ordered pairs, or the incidence matrix corresponding to such a relation. In addition, there are tools for constructing new posets from old: product and sum, subposet, distributive lattice of order ideals, etc. Once a poset has been generated (“built”), a large number of additional tools are available for studying its combina- torial structure. These existing tools reflect the authors' combinatorial interests: enumerative invariants, chains, multichains, antichains, rank generating functions, Mobius functions, linear extensions, and related topics. A good reference for these topics is Richard Stanley's book, Enumerative Combinatorics V1 (Wadsworth, 1986). About this documentation: our intention has not been to write a "users manual", but rather to give an informal demonstration of principal features. The documentation consists of two segments: Part 1 (this notebook) illustrates the basic commands by applying them to some well-known combinatorial families of posets. Part 2 (a second notebook) contains more complicated examples and illustrations, as well as “case studies” illustrating applications to combinatorial problems of interest. For a complete list of commands, one can print out the “usage” section of the package, which contains a precise description of the syntax for each command. The package is available on the web at h t t p : / / w w w . h a v e r f o r d . e d u / m a t h / c g r e e n e . h t m l . It is available in both notebook and plain ascii text formats. The authors welcome comments, questions, bug reports, criticisms, and other suggestions. Such communications should be directed to [email protected], or to Curtis Greene, Department of Mathematics, Haverford College, Haverford PA 19041. 2. Basic Instructions Loading the package

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