A Mathematica Package for Studying Posets 

Curtis Greene
Eugenie Hunsicker
Department of Mathematics
Haverford College
July 1990

Updates:  John Dollhopf, Sam Hsiao, Curtis Greene
1992-94

  1. Introduction

       This document illustrates 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 minutes 
(on a MacIIci, for example).  Smaller posets can be generated much more 
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). 

      Our intention here is not to write a "users manual", but rather to give 
an informal demonstration of the package's principal features.  For a complete 
list of commands and syntax, the user is advised to print out the "usage" 
section of the package, which has been written with this in mind.
       
       The package is available by anonymous ftp from venus.haverford.edu in 
the directory pub/cgreene/posets. 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 
cgreene@haverford.edu, or to Curtis Greene, Department of Mathematics, 
Haverford College, Haverford PA 19041.