Alternating permutations and the th descents

A half-descent of an alternating permutation π is an event that π 2 i > π 2 i + 2 , which is defined and studied in Huber and Yee (2010) [18]. The notion of half-descents is generalized to m th descents. Bivariate generating functions for permutations classified according to the numbers of m th 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.