相关论文: A Combinatorial Interpretation for the coefficient…
Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by a pair $\lambda, \mu$ of $r$-partitions and a sign $+, -$. It is expected that there…
A collection of finite sets $\{A_1, A_2,\ldots, A_{p}\}$ is said to be a double-covering if each $a\in \cup_{k=1}^{p}A_k$ is included in exactly two sets of the collection. For fixed integers $l$ and $p$, let $\mu_{l,p}$ be the number of…
In previous work of this author it was conjectured that the sum of power sums $p_\lambda,$ for partitions $\lambda$ ranging over an interval $[(1^n), \mu]$ in reverse lexicographic order, is Schur-positive. Here we investigate this…
In this paper we study incidences for hyperbolas in $\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of…
We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Pad\'e approximation. In conjunction with Honda's proof of the…
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…
We obtain a series of lower bounds for the product set of combinatorial cubes, as well as some non--trivial upper estimates for the multiplicative energy of such sets.
Given two second order free random variables $a$ and $b$, we study the second order free cumulants of their product $ab$, their commutator $ab-ba$, and their anti-commutator $ab+ba$. Let $(\kappa_n^a)_{n\geq 1}$ and…
We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calder\'{o}n--Zygmund operators on product spaces. More…
We give an explicit combinatorial formula for some irreducible components of $GL_k\times \mathbb{S}_n$-modules of multivariate diagonal harmonics. To this end we introduce a new path combinatorial object $T_{n,s}$ allowing us to give the…
The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also…
We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of…
Let $p\in(0,1)$, $\alpha:=1/p-1$ and, for any $\tau\in [0,\infty)$, $\Phi_{p}(\tau):=\tau/(1+\tau^{1-p})$. Let $H^p(\mathbb R^n)$, $h^p(\mathbb R^n)$ and $\Lambda_{n\alpha}(\mathbb{R}^n)$ be, respectively, the Hardy space, the local Hardy…
We introduce a new approach for generating combinatorial identities and formulas by the application of Kronecker substitution to polynomial expansions within quotient rings. Our main result enables the derivation of elementary arithmetic…
We describe certain special consequences of certain elementary methods from group theory for studying the algebraic complexity of matrix multiplication, as developed by H. Cohn, C. Umans et. al. in 2003 and 2005. The measure of complexity…
In a recent paper, the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable…
This paper concerns a stochastic construction of probabilistic coherent spaces by employing novel ingredients (i) linear exponential comonads arising properly in the measure-theory (ii) continuous orthogonality between measures and…
We show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero mode of an appropriate combinatorial generating function. As an application, we obtain explicit formulas for all polynomial…
We compute the modular transformation formula of the characters for a certain family of (finitely or uncountably many) simple modules over the simple $\mathcal{N}=2$ vertex operator superalgebra of central charge…
The aim of this work is to establish congruences $\left( \operatorname{mod}p^{2}\right) $ involving the trinomial coefficients $\binom{np-1}{p-1}_{2}$ and $\binom{np-1}{\left( p-1\right)/2}_{2}$ arising from the expansion of the powers of…