Alternating permutations and the mmth descents

A half-descent of an alternating permutation ππ is an event that π2i>π2i+2π2i>π2i+2, which is defined and studied in Huber and Yee (2010) [18]. The notion of half-descents   is generalized to mmth descents. Bivariate generating functions for permutations classified according to the numbers of mmth descents and inversions are given. Connections to the results on half-descents and alternating permutations are indicated. It is also shown how to specialize the presented results to obtain some other previous work on permutations.