Related papers: Viewing counting polynomials as Hilbert functions …

Let X be a closed subscheme and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d)=min{hp(P^n,d),hp(X,d)} for every non-negative…

The oriented chromatic polynomial of a oriented graph outputs the number of oriented $k$-colourings for any input $k$. We fully classify those oriented graphs for which the oriented graph has the same chromatic polynomial as the underlying…

For a hypergraph $\mathcal H$, we consider the edge-induced and vertex-induced subhypergraph polynomials and study their relation. We use this relation to prove that both polynomials are reconstructible, and to prove a theorem relating the…

The Lidskii formula for the type $A_n$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope…

We construct $S^r$-colored knot Floer homologies and prove that they satisfy categorified recurrence relations. The associated Euler characteristic implies $q$-holonomicity of the corresponding sequence of colored Alexander polynomials, in…

The purpose of this note is to introduce a new family of quasi-symmetric functions called LLT cumulants and discuss its properties. We define LLT cumulants using the algebraic framework for conditional cumulants and we prove that the…

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with…

Khovanov introduced a bigraded cohomology theory of links whose graded Euler characteristic is the Jones polynomial. The theory was subsequently applied to the chromatic polynomial of graph, resulting in a categorification known as the…

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…

Let $\lambda$ be a general length function for modules over a Noetherian ring R. We use $\lambda$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$\lambda$. We show that the leading term $\mu$ of…

For a tropical univariate polynomial $f$ we define its tropical Hilbert function as the dimension of a tropical linear prevariety of solutions of the tropical Macauley matrix of the polynomial up to a (growing) degree. We show that the…

The interior polynomial of a directed graph is defined as the $h^*$-polynomial of the graph's (extended) root polytope, and it displays several attractive properties. Here we express its degree in terms of the minimum cardinality of a…

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called \emph{optimal polynomial approximants}. In the present article, we extend such approach…

The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the…

There have been a number of recent works on the theory of period polynomials and their zeros. In particular, zeros of period polynomials have been shown to satisfy a "Riemann Hypothesis" in both classical settings and for cohomological…

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been…

We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial…

In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be…

If $R=k[x_1,\ldots,x_n]/I$ is a graded artinian algebra, then the length of $k[x_1,\ldots,x_n]/I^s$ becomes a polynomial in $s$ of degree $n$ for large $s$. If we write this polynomial as $\sum_{i=0}^n(-1)^ie_i{s+n-i-1\choose n-i}$, then…

The independence polynomial of a hypergraph is the generating function for its independent (vertex) sets with respect to their cardinality. This article aims to discuss several recurrence relations for the independence polynomial using some…