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