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), 526.
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 halfEulerian 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 halfEulerian poset, and
when we "double" every halfEulerian poset constructed, the
facet inequalities of the resulting cone are still valid for the flag
vector of any Eulerian poset.
 Flag l basis

The vectors may be read as flag lvectors of halfEulerian
posets, or flag Lvectors of their doubles.

 Sparse f basis
 The vectors should be read as sparse fvectors of
halfEulerian
posets.
For the Eulerian version, the f_{S}entry has
to be multiplied by 2^{nS}, where n is one
less
than the rank.

II. Text and Illustrations
The document, Facets of the cone up to rank 7, lists all the
inequalities
in both sparse flag fvector form and flag Lvector form. It gives
equivalent forms of the flag fvector 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 fvector and flag Lvector 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.ps, cone.pdf.
Illustrations of those halfEulerian 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
lpos.ps