The $\gamma$-Vectors of Pascal-like Triangles Defined by Riordan Arrays

TL;DR

This paper introduces and characterizes the $b3$-matrices of Pascal-like triangles derived from Riordan arrays, revealing connections to generalized Narayana triangles and the associahedron using generating functions and continued fractions.

Contribution

It defines the $b3$-matrices for Pascal-like Riordan arrays and their reversions, extending the concept to a family of generalized Narayana triangles and linking to geometric combinatorics.

Findings

Characterization of $b3$-matrices for Pascal-like Riordan arrays

Extension to reversions of these triangles in the ordinary Riordan case

Connection to generalized Narayana triangles and the associahedron

Abstract

We define and characterize the $\gamma$-matrix associated to Pascal-like matrices that are defined by ordinary and exponential Riordan arrays. We also define and characterize the $\gamma$-matrix of the reversions of these triangles, in the case of ordinary Riordan arrays. We are led to the $\gamma$-matrices of a one-parameter family of generalized Narayana triangles. Thus these matrices generalize the matrix of $\gamma$-vectors of the associahedron. The principal tools used are the bivariate generating functions of the triangles and Jacobi continued fractions.