Related papers: (-1)-Enumerations of arrowed Gelfand-Tsetlin patte…
We shall present a ``linear algebraic'' proof (involving some calculations in the algebra of linear operators on a vector space of polynomials and some manipulations of determinants) of the formula for the enumeration of symmetric…
By rewriting the famous hook-content formula it easily follows that there are $\prod\limits_{1 \le i < j \le n} \frac{k_j - k_i + j -i}{j-i}$ semistandard tableaux of shape $(k_n,k_{n-1},...,k_1)$ with entries in $\{1,2,...,n\}$ or,…
Alternating sign trapezoids have recently been introduced as a generalisation of alternating sign triangles. Fischer established a threefold refined enumeration of alternating sign trapezoids and provided three statistics on column strict…
Zeilberger proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey, for the number of alternating sign matrices with given top row. Stroganov proved an explicit…
Alternating sign triangles (ASTs) have recently been introduced by Ayyer, Behrend and the author, and it was proven that there is the same number of ASTs with n rows as there is of nxn alternating sign matrices (ASMs). We prove a conjecture…
A Gelfand model for a semisimple algebra A over C is a complex linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a…
This work is part of a project on weight bases for the irreducible representations of semisimple Lie algebras with respect to which the representation matrices of the Chevalley generators are given by explicit formulas. In the case of sl_n,…
We study a coincidence between two enumerations governed by the same product formula, reminiscent of the Robbins numbers: the unweighted enumeration of twenty-vertex configurations on quadrangular domains with fixed west boundary, and the…
This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers…
In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of $n \times n$ alternating sign matrices with a 1 at the top of the $i$-th column. We give an alternative…
We develop a simple computational tool for $SU(3)$ analogous to Bargmann's calculus for $SU(2)$. Crucial new inputs are, (i) explicit representation of the Gelfand-Zetlin basis in terms of polynomials in four variables and positive or…
We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function,…
Let $A(n,r;3)$ be the total weight of the alternating sign matrices of order $n$ whose sole `1' of the first row is at the $r^{th}$ column and the weight of an individual matrix is $3^k$ if it has $k$ entries equal to -1. Define the…
We construct closed forms that generate with repetitions all Mersenne primes, respectively all Fermat primes, all twin-prime pairs and all Sophie Germain primes. Also, we construct closed forms that count all Mersenne primes between $0$ and…
We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for…
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model…
Halved monotone triangles are a generalisation of vertically symmetric alternating sign matrices (VSASMs). We provide a weighted enumeration of halved monotone triangles with respect to a parameter which generalises the number of $-1$s in a…
We show that there is the same number of (n,l)-alternating sign trapezoids as there is of column strict shifted plane partitions of class l-1 with at most n parts in the top row, thereby proving a result that was conjectured independently…
We prove that the boundary of the Hall-Littlewood $t$-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs.…
This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear…