相关论文: Jack polynomials and some identities for partition…

In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix.…

Partition functions for non-interacting particles are known to be symmetric functions. It is shown that powerful group-theoretical techniques can be used not only to derive these relationships, but also to significantly simplify calculation…

The paper introduce a new type of partitions where the largest part appears exactly once, and the remaining parts constitute a partition of that largest part. We derive the generating function associated with these partitions and…

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…

The Hamiltonian of the quantum Calogero-Sutherland model of $N$ identical particles on the circle with $1/r^{2}$ interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials…

We derive joint factorial moment identities for point processes with Papangelou intensities. Our proof simplifies previous approaches to related moment identities and includes the setting of Poisson point processes. Applications are given…

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack…

We prove a family of partition identities which is "dual" to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and "hypergraphs" and their proof uses…

Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…

Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series $\psi(x, y, z; t, 1+\beta)$ that might be interpreted as a continuous deformation of the generating series of rooted hypermaps. They…

Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants,…

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most…

We demonstrate that statistics for several types of set partitions are described by generating functions which appear in the theory of integrable equations.

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence…

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

We investigate the joint moments of the 2k-th power of the characteristic polynomial of random unitary matrices with the 2h-th power of the derivative of this same polynomial. We prove that for a fixed h, the moments are given by rational…

This work initiates the study of {\it orthogonal} symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach…

In the paper, we give partition-theoretic results for the coefficients of some mock theta functions and prove their congruence properties. Some recurrence relations connecting the coefficients of the mock theta functions with certain…