Abstract
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 \(\sigma \) viewed as a sequence of integers, computing the weight of \(\sigma \) involves recursively counting descents of certain subpermutations of \(\sigma \). 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.
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Acknowledgements
We would like to thank Professor Einar Steingrímsson for his support and invaluable advice on this paper. We are grateful to Roger Van Peski for his dedicated mentoring and copious feedback. We thank Professor Paul E. Gunnells for proposing this area of research and for his guidance. Finally, we thank the PROMYS program and the Clay Mathematics Institute, under which this research was made possible.
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Agrawal, A., Choi, C. & Sun, N. On Permutation Weights and q-Eulerian Polynomials. Ann. Comb. 24, 363–378 (2020). https://doi.org/10.1007/s00026-020-00493-5
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DOI: https://doi.org/10.1007/s00026-020-00493-5