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On Permutation Weights and q-Eulerian Polynomials

Agrawal, A and Choi, C and Sun, N (2020) On Permutation Weights and q-Eulerian Polynomials. In: Annals of Combinatorics, 24 (2). pp. 363-378.

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Official URL: https://dx.doi.org/10.1007/s00026-020-00493-5


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 � viewed as a sequence of integers, computing the weight of � involves recursively counting descents of certain subpermutations of �. 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 et al. (Journal of Combinatorial Theory, Series A 164:24�49, 2019), and enables the definition of the formal power series Wd(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. © 2020, Springer Nature Switzerland AG.

Item Type: Journal Article
Publication: Annals of Combinatorics
Publisher: Birkhauser
Additional Information: Copy right for this article belongs to Birkhauser
Keywords: q-Eulerian polynomials, Eulerian polynomials, permutations
Department/Centre: Division of Physical & Mathematical Sciences > Physics
Date Deposited: 13 Nov 2020 10:12
Last Modified: 13 Nov 2020 10:12
URI: http://eprints.iisc.ac.in/id/eprint/65765

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