A generalization of descent polynomials

Abstract

The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics, viz. peak polynomials and symmetric functions. We define the function dm(I,n) as a generalization of the descent polynomial and we prove that for any positive integer m, this function is a polynomial in n for sufficiently large n (similarly to the descent polynomial). We obtain an explicit formula for dm(I,n) when m is sufficiently large. We look at the coefficients of dm(I,n) in different falling factorial bases. We prove the positivity of the coefficients and discover a combinatorial interpretation for them. This result is similar to the positivity result of Diaz-Lopez et al. for the descent polynomial.