Permutation Weights and a $q$-Analogue of the Eulerian Polynomials.

Weights of permutations were originally introduced by Dugan-Glennon-Gunnells-Steingru0027imsson [2] in their study of the combinatorics of tiered trees, but are still not well understood. Given a permutation $sigma$, viewed as a string of integers via two-line notation, computing the weight of $sigma$ involves counting descents of certain substrings of $sigma$. Using the weight of a permutation, one can define a $q$-analogue of the Eulerian polynomials, $E_n(x,q)$. In this paper, we prove two results about the polynomials $E_n(x,q)$. Firstly, we prove that the coefficients of $E_n(x,q)$ stabilize as $n$ goes to infinity, which was conjectured by [2] and enables the definition of certain formal power series with interesting combinatorial properties. Secondly, we prove a recursive formula for $E_n(x,q)$, similar to the known formula for the Eulerian polynomials $E_n(x)$.