Papers › On Permutation Weights and q-Eulerian Polynomials
On Permutation Weights and q-Eulerian Polynomials
Aman Agrawal, Caroline Choi, Nathan Sun
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Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingr\'imsson (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 Eₙ(x,q) of the Eulerian polynomials. We prove two main results regarding weights of permutations and the polynomials Eₙ(x,q). First, we show that the coefficients of Eₙ(x, q) stabilize as n goes to infinity, which was conjectured by Dugan, Glennon, Gunnells, and Steingr\'imsson (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 Eₙ(x, q), similar to the known recurrence for the classical Eulerian polynomials Aₙ(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.
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