A Generalization of Descent Polynomials

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 $\mathfrak{d}^{m}(I,n)$ 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 $\mathfrak{d}^{m}(I,n)$ when $m$ is sufficiently large. We look at the coefficients of $\mathfrak{d}^{m}(I,n)$ 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.