The Legendre-Stirling numbers are the coeffi cients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejoh...

These theories introduce basic concepts and proofs about discrete summation: shifts, formal summation, falling factorials and stirling numbers. As proof of concept, a simple summation conversion is provided. 1 Stirling numbers of first and second kind theory Stirling imports Main begin 1.1 Stirling numbers of the second kind fun Stirling :: nat ⇒ nat ⇒ nat where Stirling 0 0 = 1 | Stirling 0 (S...

We classify k-Stirling permutations avoiding a set of ordered patterns of length three according to Wilf-equivalence. Moreover, we derive enumeration formulæ for all of the classes using a variety of techniques such as the kernel method, a bijection related to a classical result of Simion and Schmidt, and also structural decompositions of k-Stirling permutations via the so-called block decompos...

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. 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 obta...

We show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1+q. We extend this enumerative result via a decomposition of a new poset Π(n, k) which we call the Stirling poset of the second kind. Its rank generating function is the q-Stirling number S...

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of thes...

In this work we give a study of generalizations of Stirling permutations, a restricted class of permutations of multisets introduced by Gessel and Stanley [15]. First we give several bijections between such generalized Stirling permutations and various families of increasing trees extending the known correspondences of [20, 21]. Then we consider several permutation statistics of interest for ge...

We prove an identity for sums of products of an arbitrary fixed number of Stirling numbers of the second kind; this can be seen as a generalized convolution identity. As a consequence we obtain two polynomial identities that also involve Stirling numbers of the second kind.