相关论文: Identities for the number of standard Young tablea…

We introduce an infinite family of lower triangular matrices $\Gamma^{(s)}$, where $\gamma_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of…

Given $k \geq 2$, we show that there are at most finitely many rational numbers $x$ and $y \neq 0$ and integers $\ell \geq 2$ (with $(k,\ell) \neq (2,2)$) for which $$ x (x+1) \cdots (x+k-1) = y^\ell. $$ In particular, if we assume that…

We study the Gaussent-Littelmann formula for Hall-Littlewood polynomials and we develop combinatorial tools to describe the formula in a purely combinatorial way for type A_n, B_n and C_n. This description is in terms of Young tableaux and…

We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups S_n contain all irreducibles as their constituents. Our main result is that they contain representations…

This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(\alpha)$, where $S(\alpha)$ is the set of all integers of the form $\lfloor…

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask if…

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular…

We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012).…

We count the number of alignments of $N \ge 1$ sequences when match-up types are from a specified set $S\subseteq \mathbb{N}^N$. Equivalently, we count the number of nonnegative integer matrices whose rows sum to a given fixed vector and…

For a semisimple Lie algebra g the orbit method attempts to assign representations of g to (coadjoint) orbits in g*. Orbital varieties are particular Lagrangian subvarieties of such orbits leading to highest weight representations of g. In…

Let $b^{k}_{\ell,m}(n)$ denotes the number of $k-$colored partitions of $n$ into parts that are not multiples of $\ell$ or $m$. We establish several congruence relations for $b_{\ell,m}(n)$. For instance, for any nonnegative integer $n$…

A new explicit closed-form formula for the multivariate $(n, k)$th partial Bell polynomial $B_{n,k} (x_1, x_2, ..., x_{n - k + 1})$ is deduced. The formula involves multiple summations and makes it possible, for the first time, to easily…

There are $k$ kinds of length $k$ hooks with different arm length. Actually, this $k$ kinds appear uniformly in Young diagrams of size $n$. The property ``appear uniformly'' is called super symmetric. We give a combinatorial proof of the…

We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse…

We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k} \log k)$. Prior to this work, a lemma of Agol, together with the state of the art…

Let $n,k,s$ be three integers and $\beta$ be a sufficiently small positive number such that $k\geq 3$, $0<1/n\ll \beta\ll 1/k$ and $ks+k\leq n\leq (1+\beta)ks+k-2$. A $k$-graph is called non-trivial if it has no isolated vertex. In this…

The hook length formula for $d$-complete posets expresses the number of linear extensions of a $d$-complete poset $P$ in terms of hooks of $P$. It generalizes the usual hook length formula for standard Young tableaux, as well as hook length…

In this short note, we give large counterexamples to natural questions about certain order polytopes, in particular, Gelfand--Tsetlin polytopes. Several of the counterexamples are too large to be discovered via a brute-force computer…

The family of graphs of reduced words of a certain subcollection of permutations in the union $\cup_{n\geq 4}\frak{S}_{n}$ of symmetic groups is investigated. The subcollection is characterised by the hook cycle type $(n-2,1,1)$ with…

Let $\mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the "skew three-rowed strip" $\mathcal{T}_3 / (2,1,0)$ is $m_{n-1}-m_{n-3}$, a difference of two…