Journal of Integer Sequences, Vol. 27 (2024), Article 24.1.5

Generalized Eulerian Polynomials with a Nonnegative Gamma Vector

Pasquale Petrullo and Domenico Senato
Università degli Studi della Basilicata
Dipartimento di Scienze Umane
Via Nazario Sauro 85
85100 Potenza


We define a family of generalized Eulerian polynomials depending on three parameters. We prove that these polynomials have a nonnegative gamma vector, and we provide a combinatorial description of the corresponding gamma coefficients. By assigning suitable integer values to the parameters, we obtain a new expansion of the nth Eulerian polynomial over the symmetric group 𝔖n−1, a new description of the associated gamma vector, and an identity relating the derangements of 𝔖2n to the alternating permutations of 𝔖2n+1.

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(Concerned with sequences A000166 A001250 A008292.)

Received August 5 2023; revised version received August 6 2023; November 18 2023; November 20 2023. Published in Journal of Integer Sequences, January 13 2024.

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