Colored Multipermutations and a Combinatorial Generalization of Worpitzky's Identity

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.