Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions

Abstract

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions.We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers. * This paper is part of the author's Ph.D. thesis written under the direction of Prof. F. Brenti at the Univ. "la Sapienza" of Rome, Italy.

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

“…In this paper, we obtain explicit expressions for any positive integer power of the q-differential operator L q given in (1.2) and the key to achieve this is via the introduction of a new set of numbers, that we call as q-Jacobi-Stirling numbers, to which we obtain several properties as well as a combinatorial interpretation. The results here obtained are the q-version to those in [26] (and also in [7], [8]), as well as to those in [2], [5], [11], [12], [31], [32], since we provide here combinatorial interpretations to the arisen coefficients and eigenvalues. This study has the merit of addressing all the q-classical polynomial sequences as whole in a coherent framework that brings together generalisations of the q-differential equation (1.2) and associated combinatorial interpretations.…”

“…where α represents a complex number. Since their introduction, several properties of the Jacobi-Stirling numbers and its companions including combinatorial interpretations have been established, see [2], [5], [11], [12], [24], [32].…”

“…Remark 2.4 By letting q → 1, the pair Jc k n (z; q), JS k n (z; q) 0 k n reduces to the Jacobi-Stirling numbers in [7], [11], [32] and the relation (2.12) reduces to [7, (4.4)].…”

Asymptotics of Stirling and Chebyshev‐Stirling Numbers of the Second Kind

“…Here, due to the factorization in (25), the entries are row-wise independent with distributions given by P(X nν = 1) = p nν , P(X nν = 0) = 1 − p nν (27) with numbers p nν ∈ [0, 1]. In [16], an asymptotic expansion for p(n, j) := P(S n = j) (28) has been derived, where S n := n ν=1 X nν (29) denote the row sums in the scheme (26). This has been done by a modification of a standard local central limit theorem for "simple sums" n ν=1 X ν , where each component X ν has a lattice distribution [30].…”

On the asymptotic normality of the Legendre–Stirling numbers of the second kind

“…The Legendre differential operator corresponds to the case α = β = 0, i.e., γ = 1, and hence the numbers n j 1 are called Legendre-Stirling numbers of the second kind. Besides the already mentioned original field of differential equations during the past decade the Jacobi-Stirling numbers received considerable attention especially in combinatorics and graph theory, see, e.g., [1], [2], [3], [6], [7], [9], [15], [16], [17], [21], [22], [23]. Among the n j γ 's the Legendre-Stirling numbers n j 1…”