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Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics.…
In this paper, by using some families of special numbers and polynomials with their generating functions, we give various properties of these numbers and polynomials. These numbers are related to the well-known numbers and polynomials,…
Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric kind is a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the…
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
We give an explicit raising operator formula for the modified Macdonald polynomials $\tilde{H}_{\mu }(X;q,t)$, which follows from our recent formula for $\nabla$ on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified…
This paper surveys eight classes of polynomials associated with $A$-type and $BC$-type root systems: Jack, Jacobi, Macdonald and Koornwinder polynomials and interpolation (or shifted) Jack and Macdonald polynomials and their $BC$-type…
A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for…
We investigate the connections between various noncommutative analogues of Hall-Littlewood and Macdonald polynomials, and define some new families of noncommutative symmetric functions depending on two sequences of parameters.
In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,…
This is a paper about $c$-functions and Macdonald polynomials. There are $c$-function formulas for $E$-expansions of $P_\lambda$ and $A_{\lambda+\rho}$, principal specializations of $P_\lambda$ and $E_\mu$, for Macdonald's constant term…
We give a concise direct proof of the orthogonality of interpolation Macdonald polynomials with respect to the Fourier pairing and briefly discuss some immediate applications of this orthogonality, such as the symmetry of the Fourier…
We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared…
We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials,…
We introduce Macdonald polynomials indexed by $n$-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials…
We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of…
Borel's triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel's triangle. We present various combinatorial interpretations of Borel's triangle in…
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…