论文标题
通过下降位置的平等枚举列出
Enumeration of permutations by the parity of descent positions
论文作者
论文摘要
注意到，最近的血统多项式的一些变化是卡利兹和斯科维尔的四变量多项式的特殊案例，通过血统和上升的位置列举列出，我们证明了$ q $ q $ q $ - carlitz-coville生成函数的符合imume blogie and type blogie fief inversie blogie of type blogie of type blogie of type blogue of type beriumer的符合符号，并desecnt和上升位置。作为我们配方的副产品，我们获得了Chebikin的配方奶粉的$ Q $ Analogue，用于交替的下降多项式，这是Sun对她的双变量欧拉群岛多种主管的伽马积极性的另一种证明，而后者是Petersen的petersen gama positivity the the dyperian byn of typle b eulerian polynian typle bynian typerian fornogia。
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.
