Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles
Robert S. Maier
TL;DR
The paper develops a unified, parametric framework for GKP-type triangular recurrences, introducing a six-parameter GKP notation and an $S_3$-group of row-wise transformations that act on both the triangles and their EGFs. It applies the method of characteristics to derive a Gauss ${}_2F_1$-based implicit representation of the bivariate EGF, and identifies three major solvable families: generalized Stirling–Eulerian (A), generalized Narayana (B), and generalized secant–tangent (C) triangles. Within this framework, it defines and analyzes generalized Stirling numbers $S_{n,k}(a,b;r)$ and generalized Eulerian numbers $E_{n,k}(a,b;c_0,c_ fty)$, establishing rank-1 formulas, connection-coefficient interpretations, and widespread identities (contiguity, binomial transforms, and Worpitzky-type formulas). The work also develops Narayana and secant–tangent generalizations, providing closed forms in notable subcases and linking to combinatorial structures such as noncrossing partitions and polytope f- and h-vectors, with numerous connections to Riordan arrays and hypergeometric representations. Overall, the paper offers a cohesive analytic and algebraic framework for broad classes of combinatorial triangles and their interrelations.
Abstract
Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.