Matrix products of binomial coefficients and unsigned Stirling numbers

TL;DR

This paper investigates sums involving binomial coefficients and unsigned Stirling numbers, revealing their combinatorial properties, recurrence relations, and polynomial basis interpretations, with some sums expressible in closed form.

Contribution

It provides new insights into the structure and relationships of sums of binomial coefficients and Stirling numbers, including closed forms and combinatorial interpretations.

Findings

Some sums can be expressed in closed form.

Identifies Pascal-like recurrence relations.

Establishes inverse relations with signed versions.

Abstract

We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common features: combinatorial interpretations, Pascal-like recurrences, inverse relations with their signed versions, and interpretations as coefficients of change between polynomial bases.