Some determinants of path generating functions

TL;DR

This paper evaluates determinants of matrices with entries based on path generating functions, leading to new and known results for combinatorial numbers like Catalan, ballot, and Motzkin numbers.

Contribution

It introduces four families of determinant evaluations involving path generating functions, unifying and extending previous combinatorial determinant results.

Findings

Derived new determinant formulas for path generating functions.

Unified existing results for Catalan, ballot, and Motzkin numbers.

Provided corollaries with numerous combinatorial number evaluations.

Abstract

We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers - most notably of Catalan, ballot, and of Motzkin numbers - that appeared previously in the literature.