2022/09/30 by Qiongqiong Pan, Jiang Zeng, Pan, Qiongqiong +1 · 1 citation
Mathematics · Agricultural and Biological Sciences · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Botanical Research and Chemistry
paper · pdf · doi:10.48550/arxiv.2209.15302
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.