218 research outputs found

### Ehrhart f*-coefficients of polytopal complexes are non-negative integers

The Ehrhart polynomial $L_P$ of an integral polytope $P$ counts the number of
integer points in integral dilates of $P$. Ehrhart polynomials of polytopes are
often described in terms of their Ehrhart $h^*$-vector (aka Ehrhart
$\delta$-vector), which is the vector of coefficients of $L_P$ with respect to
a certain binomial basis and which coincides with the $h$-vector of a regular
unimodular triangulation of $P$ (if one exists). One important result by
Stanley about $h^*$-vectors of polytopes is that their entries are always
non-negative. However, recent combinatorial applications of Ehrhart theory give
rise to polytopal complexes with $h^*$-vectors that have negative entries.
In this article we introduce the Ehrhart $f^*$-vector of polytopes or, more
generally, of polytopal complexes $K$. These are again coefficient vectors of
$L_K$ with respect to a certain binomial basis of the space of polynomials and
they have the property that the $f^*$-vector of a unimodular simplicial complex
coincides with its $f$-vector. The main result of this article is a counting
interpretation for the $f^*$-coefficients which implies that $f^*$-coefficients
of integral polytopal complexes are always non-negative integers. This holds
even if the polytopal complex does not have a unimodular triangulation and if
its $h^*$-vector does have negative entries. Our main technical tool is a new
partition of the set of lattice points in a simplicial cone into discrete
cones. Further results include a complete characterization of Ehrhart
polynomials of integral partial polytopal complexes and a non-negativity
theorem for the $f^*$-vectors of rational polytopal complexes.Comment: 19 pages, 1 figur

### A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins

The main result of this paper is a bijective proof showing that the
generating function for partitions with bounded differences between largest and
smallest part is a rational function. This result is similar to the closely
related case of partitions with fixed differences between largest and smallest
parts which has recently been studied through analytic methods by Andrews,
Beck, and Robbins. Our approach is geometric: We model partitions with bounded
differences as lattice points in an infinite union of polyhedral cones.
Surprisingly, this infinite union tiles a single simplicial cone. This
construction then leads to a bijection that can be interpreted on a purely
combinatorial level.Comment: 12 pages, 5 figure

### Bounds on the Coefficients of Tension and Flow Polynomials

The goal of this article is to obtain bounds on the coefficients of modular
and integral flow and tension polynomials of graphs. To this end we make use of
the fact that these polynomials can be realized as Ehrhart polynomials of
inside-out polytopes. Inside-out polytopes come with an associated relative
polytopal complex and, for a wide class of inside-out polytopes, we show that
this complex has a convex ear decomposition. This leads to the desired bounds
on the coefficients of these polynomials.Comment: 16 page

### Enumerating Colorings, Tensions and Flows in Cell Complexes

We study quasipolynomials enumerating proper colorings, nowhere-zero
tensions, and nowhere-zero flows in an arbitrary CW-complex $X$, generalizing
the chromatic, tension and flow polynomials of a graph. Our colorings, tensions
and flows may be either modular (with values in $\mathbb{Z}/k\mathbb{Z}$ for
some $k$) or integral (with values in $\{-k+1,\dots,k-1\}$). We obtain
deletion-contraction recurrences and closed formulas for the chromatic, tension
and flow quasipolynomials, assuming certain unimodularity conditions. We use
geometric methods, specifically Ehrhart theory and inside-out polytopes, to
obtain reciprocity theorems for all of the aforementioned quasipolynomials,
giving combinatorial interpretations of their values at negative integers as
well as formulas for the numbers of acyclic and totally cyclic orientations of
$X$.Comment: 28 pages, 3 figures. Final version, to appear in J. Combin. Theory
Series

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