Colored Multipermutations and a Combinatorial Generalization of Worpitzky's Identity

TL;DR

This paper introduces a combinatorial interpretation for generalized Worpitzky identity numbers using colored multipermutations, providing new insights and proofs for related combinatorial identities.

Contribution

It defines descents in colored multipermutations and shows these count the numbers in the generalized Worpitzky identity, offering a novel combinatorial perspective.

Findings

A combinatorial interpretation of $A_{a,b,r}(p,i)$ as colored multipermutations with weak descents.

Proofs of identities involving $A_{a,b,r}(p,i)$ using combinatorial methods.

Extension of Worpitzky's identity to a broader combinatorial context.

Abstract

Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$ Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a generalized Worpitzky identity $$\binom{an+b}{r}^p = \sum_{i=0}^{rp} A_{a,b,r}(p,i) \binom{n+rp-i}{rp},$$ and asked whether there is a combinatorial interpretation of the numbers $A_{a,b,r}(p,i)$. We provide such a combinatorial interpretation by defining a notion of descents in colored multipermutations, and then proving that $A_{a,b,r}(p,i)$ is equal to the number of colored multipermutations of $\{1^r, 2^r, \ldots, p^r\}$ with $a$ colors and $i$ weak descents. We use this to give combinatorial proofs of several identities involving $A_{a,b,r}(p,i)$, including the aforementioned generalized Worpitzky identity.