The cone of flag vectors of Eulerian posets up to rank 7

This page contains documentation for our paper
"Flag Vectors of Eulerian Partially Ordered Sets", by Margaret Bayer and Gábor Hetyei, European Journal of Combinatorics 22 (2001), 5-26.

I. PORTA files

PORTA, written by Thomas Christoph and Andreas Löbel, is a collection of routines for analyzing polytopes and polyhedra. After constructing the extreme rays as limits of normalized flag vectors of half-Eulerian posets, we used this program to obtain the facet inequalities of the resulting cones. We then verified that these inequalities hold for the flag vector of any half-Eulerian poset, and when we "double" every half-Eulerian poset constructed, the facet inequalities of the resulting cone are still valid for the flag vector of any Eulerian poset.
  1. Flag l basis

    The vectors may be read as flag l-vectors of half-Eulerian posets, or flag L-vectors of their doubles.

    Rank: 3 4 5 6 7
    Porta input files:
    (including the output as a comment)
    rank3l.poi rank4l.poi rank5l.poi rank6l.poi rank7l.poi
    Explanation: rank3l.txt rank4l.txt rank5l.txt rank6l.txt rank7l.txt

  2. Sparse f basis

    The vectors should be read as sparse f-vectors of half-Eulerian posets.
    For the Eulerian version, the fS-entry has to be multiplied by 2n-|S|, where n is one less than the rank.

    Rank: 3 4 5 6 7
    Porta input files:
    (Including the output as a comment)
    rank3f.poi rank4f.poi rank5f.poi rank6f.poi rank7f.poi
    Explanation: rank3f.txt rank4f.txt rank5f.txt rank6f.txt rank7f.txt

II. Text and Illustrations

The document, Facets of the cone up to rank 7, lists all the inequalities in both sparse flag f-vector form and flag L-vector form. It gives equivalent forms of the flag f-vector inequalities so that their validity can be checked using Proposition 3.1 and Theorem 3.2 of the paper, Flag vectors of Eulerian partially ordered sets. It also shows how to match up the flag f-vector and flag L-vector forms. (This is necessary because PORTA produced the inequalities in different orders.) To get this document, click on the file cone in the format that you prefer: cone.tex, cone.dvi,, cone.pdf.

Illustrations of those half-Eulerian limits posets of rank 7 which do not arise as amplifying an even system of intervals are available as a LaTex file lpos.tex, a DVI file lpos.dvi, or a PS file

Copyright © 2000 Margaret Bayer and Gábor Hetyei bayer "at"
Last update: May 10, 2005.