A new family of generalized Stirling and Bell numbers is introduced by considering powers (V U) n of the noncommuting variables U, V satisfying U V = V U + hV s. The case s = 0 (and h = 1) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit… (More)
The generalized Stirling numbers S s;h (n, k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n, k; α, β, r) considered by Hsu and Shiue. From this relation, several properties of S s;h (n, k) and the associated Bell numbers B s;h (n) and Bell polynomials B s;h|n (x) are… (More)
The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given.
As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to n-generalized Fibonacci numbers is established. Several other classes of differential equations (systems of first order ,… (More)
The generalized Stirling numbers introduced recently [11, 12] are considered in detail for the particular case s = 2 corresponding to the meromorphic Weyl algebra. A combinatorial interpretation in terms of perfect matchings is given for these meromorphic Stirling numbers and the connection to Bessel functions is discussed. Furthermore, two related… (More)
We discuss the generalized Touchard polynomials introduced recently by Dattoli et al. as well as their extension to negative order introduced by the authors with operationial methods. The connection to generalized Stirling and Bell numbers is elucidated and analogs to Burchnall's identity are derived. A recursion relation for the generalized Touchard… (More)
Suppose π = π 1 π 2 · · · πn is a partition of size n, represented in its canonical sequential form. We show that the number of partitions of size n so represented having no two adjacent letters the same and avoiding a single pattern of length five is given by the Catalan number C n−1 in six particular instances. In addition to supplying apparently new… (More)
Recently the vertex Padmakar–Ivan (PI v) index of a graph G was introduced as the sum over all edges e = uv of G of the number of vertices which are not equidistant to the vertices u and v. In this paper the vertex PI index and Szeged index of bridge graphs are determined. Using these formulas, the vertex PI indices and Szeged indices of several graphs are… (More)