Dyck paths, Motzkin paths, and the binomial transform

We study the moments of orthogonal polynomial sequences (OPS) arising from tridiagonal matrices. We obtain combinatorial information about the sequence of moments of some OPS in terms of Motzkin and Dyck paths, and also in terms of the binomial transform. We then introduce an equivalence relation on the set of Dyck paths and some operations on them. We determine a formula for the cardinality of those equivalence classes, and use this information to obtain a combinatorial formula for the number of Dyck and Motzkin paths of a fixed length

The Rogers--Ramanujan recursion and intertwining operators

We use vertex operator algebras and intertwining operators to study certain substructures of standard $A_1^{(1)}$--modules, allowing us to conceptually obtain the classical Rogers--Ramanujan recursion. As a consequence we recover Feigin-Stoyanovsky's character formulas for the principal subspaces of the level 1 standard $A_1^{(1)}$--modules.Comment: minor change

A generalization of the "probléme des rencontres"

In this paper, we study a generalization of the classical \emph{probl\'eme des rencontres} (\emph{problem of coincidences}), consisting in the enumeration of all permutations \pi \in \SS_n with $k$ fixed points, and, in particular, in the enumeration of all permutations \pi \in \SS_n with no fixed points (derangements). Specifically, we study this problem for the permutations of the $n+m$ symbols $1$, $2$, \ldots, $n$, $v_1$, $v_2$, \ldots, $v_m$, where $v_i \not\in\{1,2,\ldots,n\}$ for every $i=1,2,\ldots,m$. In this way, we obtain a generalization of the derangement numbers, the rencontres numbers and the rencontres polynomials. For these numbers and polynomials, we obtain the exponential generating series, some recurrences and representations, and several combinatorial identities. Moreover, we obtain the expectation and the variance of the number of fixed points in a random permutation of the considered kind. Finally, we obtain some asymptotic formulas for the generalized rencontres numbers and the generalized derangement numbers

New partition identities from (C^{(1)}_ell)-modules

In this paper we conjecture combinatorial Rogers-Ramanu­jan type colored partition identities related to standard representations of the affine Lie algebra of type (C^{(1)}_ell), (ellgeq2), and we conjecture similar colored partition identities with no obvious connection to representation theory of affine Lie algebras

On some theorems of Hirschhorn

It is found that four theorems of Hirschhorn [Hirschhorn, M. D. (1979). Some partition theorems of the Rogers-Ramanujan type. J. Comb. Th. A 27:33-37] correspond to standard representations of an affine Lie algebra. Using this correspondence two more theorems of the same type are found. It turns out that a classical result of Euler also appears in this context