Related papers: Irrational proofs for three theorems of Stanley
We give a short, self-contained proof of Stanley's reciprocity theorem for a rational cone K \subset R^d. Namely, let sigma_K (x) = sum_{m \in K \cap Z^d} x^m. Then sigma_K (x) and sigma_int(K) (x) are rational functions which satisfy the…
The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition…
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its…
We generalize R. P. Stanley's celebrated theorem that the $h^\ast$-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to…
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly…
The Ehrhart quasipolynomial of a rational polytope $\mathsf{P}$ encodes fundamental arithmetic data of $\mathsf{P}$, namely, the number of integer lattice points in positive integral dilates of $\mathsf{P}$. Ehrhart quasipolynomials were…
Stanley's non-negativity theorem is at the heart of many of the results in Ehrhart theory. In this paper, we analyze the root behavior of general polynomials satisfying the conditions of Stanley's theorem and compare this to the known root…
We give a new proof for a theorem of Ehrhart regarding the quasi-polynomiality of the function that counts the number of integer points in the integral dilates of a rational polytope. The proof involves a geometric bijection,…
For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an intimate relation between these two counting functions. A…
A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ --- that is, a "polynomial" in which the coefficients are…
We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…
A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a…
We express the generating function for lattice points in a rational polyhedral cone with a simplicial subdivision in terms of multivariate analogues of the h-polynomials of the subdivision and "local contributions" of the links of its…
In a recent paper, Cristofaro-Gardiner--Li--Stanley [CGLS15] constructed examples of irrational triangles whose Ehrhart functions (i.e. lattice-point count) are polynomials when restricted to positive integer dilation factors. This is very…
If $P$ is a lattice polytope (i.e., $P$ is the convex hull of finitely many integer points in $\mathbb{R}^d$) of dimension $d$, Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|nP \cap \mathbb{Z}^d|$ is a…
Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a…
Graph polytopes arising from vertex-weighted graphs were first introduced by B\'ona, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is…
If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap…
In this paper we study the generating function f(t) for the sequence of the moments \int_{\gamma}P^i(z)q(z)d z, i\geq 0, where P(z),q(z) are rational functions of one complex variable and \gamma is a curve in C. We calculate an analytical…
We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expository note contains a new proof of Brion's Formula using…