Related papers: Counting subspaces of a finite vector space

In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients.…

In this article, we present a short, non-exhaustive study of an important and well-known property of combinatorial sequences - unimodality. We shall have a look at a sample of classical results on unimodality and related properties, and…

We introduce the subsum polynomial of a partition $\lambda=(\lambda_1, \lambda_2, \ldots, \lambda_k)$ defined by $\mathrm{sp}(\lambda, x)=\prod_{i=1}^k(1+x^{\lambda_i})$. We study the sum of reciprocals of $\mathrm{sp}(\lambda, x)$ over all…

Set partitions and permutations with restrictions on the size of the blocks and cycles are important combinatorial sequences. Counting these objects lead to the sequences generalizing the classical Stirling and Bell numbers. The main focus…

Gaussian blur is a commonly-used method to filter image data. This paper introduces the collapsing sum, a new operator on matrices that provides a combinatorial interpretation of Gaussian blur. We study the combinatorial properties of this…

This is an expository article on the Poisson binomial distribution. We review lesser known results and recent progress on this topic, including geometry of polynomials and distribution learning. We also provide examples to illustrate the…

We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those…

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with…

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…

We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating…

A multivariate Gauss-Lucas theorem is proved, sharpening and generalizing previous results on this topic. The theorem is stated in terms of a seemingly new notion of convexity. Applications to multivariate stable polynomials are given.

Modelling spatio-temporal processes has become an important issue in current research. Since Gaussian processes are essentially determined by their second order structure, broad classes of covariance functions are of interest. Here, a new…

In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the…

A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson…

Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to…

Let $G$ be a group and let $X$ be a transitive $G$-space. We classify the subsets of $X$ with respect to a translation invariant ideal $\mathcal{J}$ in the Boolean algebra of all subsets of $X$, introduce and apply the relative…

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined…

The $(q,r)$-Whitney numbers were recently defined in terms of the $q$-Boson operators, and several combinatorial properties which appear to be $q$-analogues of similar properties were studied. In this paper, we obtain elementary and…

We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we…

This survey provides an exposition of a suite of techniques based on the theory of polynomials, collectively referred to as polynomial methods, which have recently been applied to address several challenging problems in statistical…