On Permutation Weights and q -Eulerian Polynomials

Weights of permutations were originally introduced by Dugan et al. ( Journal of Combinatorial Theory, Series A 164:24–49, 2019) in their study of the combinatorics of tiered trees. Given a permutation σ viewed as a sequence of integers, computing the weight of σ involves recursively counting descents of certain subpermutations of σ . Using this weight function, one can define a q -analog E_n(x,q) of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials E_n(x,q) . First, we show that the coefficients of E_n(x, q) stabilize as n goes to infinity, which was conjectured by Dugan et al. ( Journal of Combinatorial Theory, Series A 164:24–49, 2019), and enables the definition of the formal power series W_d(t) , which has interesting combinatorial properties. Second, we derive a recurrence relation for E_n(x, q) , similar to the known recurrence for the classical Eulerian polynomials A_n(x) . Finally, we give a recursive formula for the numbers of certain integer partitions and, from this, conjecture a recursive formula for the stabilized coefficients mentioned above.