Related papers: Signed Mahonian Polynomials on Colored Derangement…

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2,C), for any possible holonomy around the puncture. We follow the geometric technique introduced…

Given a graph $G$, the $k$-coloring graph $\mathcal{C}_k(G)$ is constructed by selecting proper $k$-colorings of $G$ as vertices, with an edge between two colorings if they differ in the color of exactly one vertex. The number of vertices…

We study the combinatorial equivalence of separable elements in types $A$ and $B$. A bijection is constructed from the set of separable permutations in the symmetric group $S_{n+1}$ to the set of separable signed permutations in the…

MacMahon's classic theorem states that the 'length' and 'major index' statistics are equidistributed on the symmetric group S_n. By defining natural analogues or generalizations of those statistics, similar equidistribution results have…

The article presents results on the well-known problem concerning the structure of integer polynomials $p_n(z; x, y)$, which define multiplication laws in $n$-valued groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. We…

In this paper, we introduce the concept of the over-Mahonian number, which counts the overlined permutations of length $n$ with $k$ inversions, allowing the first elements associated with the inversions to be independently overlined or not.…

Given a rank $n$ irreducible finite reflection group $W$, the $W$-invariant polynomial functions defined in ${\mathbb R}^n$ can be written as polynomials of $n$ algebraically independent homogeneous polynomial functions,…

The edge-Szeged index of a graph $G$ is defined as $Sz_e(G) = \sum_{e=uv \in E(G)}m_u(e)m_v(e)$, where $m_u(e)$ denotes the number of edges of $G$ whose distance to $u$ is smaller than the distance to $v$ and $m_v(e)$ denotes the number of…

With quantum groups $U_q(su_n)$ taken as classifying symmetries for hadrons of $n$ flavors, we calculate within irreducible representation $D^+_{12}(p-1,p-3,p-4;p,p-2)$ ($p \in {\bf Z}$) of 'dynamical' quantum group $U_q(u_{4,1})$ the…

We prove an interesting fact describing the location of the roots of the generating polynomials of the numbers of derangements of length $n$, counted by their number of cycles. We then use this result to prove that if $k$ is the number of…

The $\sigma$-polynomial is given by $\sigma(G,x) = \sum_{i=\chi(G)}^{n} a_{i}(G)\, x^{i}$, where $a_{i}(G)$ is the number of partitions of the vertices of $G$ into $i$ nonempty independent sets. These polynomials are closely related to…

For any connected graph $G$, let $P(G,m)$ and $P_{DP}(G,m)$ denote the chromatic polynomial and DP color function of $G$, respectively. It is known that $P_{DP}(G,m)\le P(G,m)$ holds for every positive integer $m$. Let $DP_\approx$ (resp.…

We present a general framework for constructing polynomial integrable systems on linearizations of Poisson varieties that admit log-canonical systems. Our construction is in particular applicable to Poisson varieties with compatible cluster…

Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a moderate asymmetric expander in the sense that $|P(A,B)|…

Asymptotic expansions of Gaussian integrals may often be interpreted as generating functions for certain combinatorial objects (graphs with additional data). In this article we discuss a general approach to all such cases using colored…

In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of…

Most results on the value sets $V_f$ of polynomials $f \in \mathbb{F}_q[x]$ relate the cardinality $|V_f|$ to the degree of $f$. In particular, the structure of the spectrum of the class of polynomials of a fixed degree $d$ is rather well…

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure…

Fix an integer $d \geq 2$. The space $\mathcal{P}_{d}$ of polynomial maps of degree $d$ modulo conjugation by affine transformations is naturally an affine variety over $\mathbb{Q}$ of dimension $d -1$. For each integer $P \geq 1$, the…