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Noticing that some recent variations of descent polynomials are special casesof Carlitz and Scoville's four-variable polynomials, which enumeratepermutations by the parity of descent and ascent positions, we prove a$q$-analogue of Carlitz-Scoville's generating function by counting theinversion number and a type B analogue by enumerating the signed permutationswith respect to the parity of desecnt and ascent positions. As a by-product ofour formulas, we obtain a $q$-analogue of Chebikin's formula for alternatingdescent polynomials, an alternative proof of Sun's gamma-positivity of herbivariate Eulerian polynomials and a type B analogue, the latter refinesPetersen's gamma-positivity of the type B Eulerian polynomials.

Enumeration of permutations by the parity of descent positions