Colored multipermutations and a combinatorial generalization of Worpitzky's identity

Worpitzky's identity, first presented in 1883, expresses n(p) in terms of the Eulerian numbers and binomial coefficients: n(p) = -Sigma(p-1)(i=0) < p i > (n+1 p). Pita-Ruiz recently defined numbers A(a,b,r)(p, i) implicitly to satisfy a generalized Worpitzky identity (an+b r)(p) = Sigma(rp)(i=0) A(a,b,r)(p, i)(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), ..., 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.