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We devise an explicit method for computing combinatorial formulae for Hadamard products of certain rational generating functions. The latter arise naturally when studying so-called ask zeta functions of direct sums of modules of matrices or…

We study two new families of symmetric functions arising from a species-theoretic construction motivated by cycle structure. For each partition of $n$, we define two combinatorial species that decompose into molecules indexed by the same…

We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions,…

We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field…

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for…

This is the first of two articles devoted to a exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit…

We derive asymptotic formulae for the coefficients of bivariate generating functions with algebraic and logarithmic factors. Logarithms appear when encoding cycles of combinatorial objects, and also implicitly when objects can be broken…

Summation formulas are obtained for products of associated Lagurre polynomials by means of the Green's function K for the Hamiltonian H = -{d^2\over dx^2} + x^2 + Ax^{-2}, A > 0. K is constructed by an application of a Mercer type theorem…

We define dual equivalence for any collection of combinatorial objects endowed with a descent set, and we show that giving a dual equivalence establishes the symmetry and Schur positivity of the quasi-symmetric generating function. We give…

A recurrent theme in functional analysis is the interplay between the theory of positive definite functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. This central theme is motivated by…

We present a single operation for constructing skew diagrams whose corresponding skew Schur functions are equal. This combinatorial operation naturally generalises and unifies all results of this type to date. Moreover, our operation…

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for…

We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce…

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a…

We define a new basis of quasisymmetric functions, the row-strict dual immaculate functions, as the generating function of a particular set of tableaux. We establish that this definition gives a function that can also be obtained by…

We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…

We give a combinatorial expansion of a Schubert homology class in the affine Grassmannian Gr_{SL_k} into Schubert homology classes in Gr_{SL_{k+1}}. This is achieved by studying the combinatorics of a new class of partitions called…

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of "mixed" products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine…

We introduce the arrow product, a systematic generating function technique for directed graph enumeration. It provides short proofs for previous results of Gessel on the number of directed acyclic graphs and of Liskovets, Robinson and…

Two methods of constructing 2D Toda $\tau$-functions that are generating functions for certain geometrical invariants of a combinatorial nature are related. The first involves generation of paths in the Cayley graph of the symmetric group…