Marked mesh patterns are a very general type of permutation pattern. We examine a particular marked mesh pattern originally defined by Kitaev and Remmel, and show that its generating function is described by the r-Stirling numbers. We examine some ramifications of various properties of the r-Stirling numbers for this generating function, and find (seemingly new) formulas for the r-Stirling numb...

We establish an extension of Spivey’s Bell number formula and its associated Bell polynomial extension by using Hsu-Shiue’s generalized Stirling numbers. By means of the extension of Spivey’s Bell number formula we also extend Gould-Quaintance’s new Bell number formulas.

Ferrers graphs are bipartite graphs that correspond naturally to Ferrers shapes. In this paper, we determine the homotopy type of the boolean complex of Ferrers graphs. Previous work of the last two authors shows that the boolean complex of a graph is a wedge of spheres of maximal dimension. Thus the homotopy type of this complex depends entirely on the number of spheres in the wedge sum, calle...

We study a particular number pyramid bn,k,l that relates the binomial, Deleham, Eulerian, MacMahon-type and Stirling number triangles. The numbers bn,k,l are generated by a function Bc(x, y, t), c ∈ C, that appears in the calculation of derivatives of a class of functions whose derivatives can be expressed as polynomials in the function itself or a related function. Based on the properties of t...

We present new short proofs for both Stirling’s formula and Stirling’s formula for the Gamma function. Our approach in the …rst case relies upon analysis of Wallis’formula, while the second result follows from the logconvexity property of the Gamma function. The well known formula of Stirling asserts that (1) n! p 2 ne n as n!1; in the sense that the ratio of the two sides tends to 1. This prov...

We derive a family of labeled, undirected graphs from the Stirling table of the rst kind and investigate properties of these graphs as a basis for multiprocessor interconnection networks. The diameter of a Stirling network with n nodes is dlog2(n + 1)e 1, the average distance is less than 10/3, and the number of links is O(n1:59). Stirling networks can be inductively speci ed with incrementabil...

Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli Euler numbers.

Peter Kovesi Eun-Jung Holden Jason C. Wong Centre for Exploration Targeting Centre for Exploration Targeting Centre for Exploration Targeting The University of Western Australia The University of Western Australia The University of Western Australia 35 Stirling Highway. Crawley WA. 6009 35 Stirling Highway. Crawley WA. 6009 35 Stirling Highway. Crawley WA. 6009 [email protected] eun-jung....

We employ the generalized factorials to define a Stirling-type pair {s n, k;α,β, r , S n, k;α,β, r } which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The g...

In this paper we provide constructive proofs that the following three statistics are equidistributed: the number of ascent plateaus of Stirling permutations of order n, a weighted variant of the number of excedances in permutations of length n and the number of blocks with even maximal elements in perfect matchings of the set {1, 2, 3, . . . , 2n}.