Enumeration of permutations by the parity of descent positions

Qiongqiong Pan; Jiang Zeng

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arXiv · 2209.15302

Enumeration of permutations by the parity of descent positions

Qiongqiong Pan·Jiang Zeng

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Abstract

Noticing that some recent variations of descent polynomials are special cases of Carlitz and Scoville's four-variable polynomials, which enumerate permutations by the parity of descent and ascent positions, we prove a $q$-analogue of Carlitz-Scoville's generating function by counting the inversion number and a type B analogue by enumerating the signed permutations with respect to the parity of desecnt and ascent positions. As a by-product of our formulas, we obtain a $q$-analogue of Chebikin's formula for alternating descent polynomials, an alternative proof of Sun's gamma-positivity of her bivariate Eulerian polynomials and a type B analogue, the latter refines Petersen's gamma-positivity of the type B Eulerian polynomials.

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Qiongqiong Pan, Jiang Zeng. 2023-06-13. Enumeration of permutations by the parity of descent positions. https://arxiv.org/abs/2209.15302

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