Combinatorics of the group of parity alternating permutations

A permutation is called parity alternating if its entries assume even and odd integers alternately. Parity alternating permutations form a subgroup of the symmetric group. This paper deals with the number of permutations classified by the ascent number. These numbers have a close relationship to signed Eulerian numbers. The present approach is based on a study of permutations that are not parity alternating. It is proved that the number of even permutations is equal to that of odd ones in it. Hence signed Eulerian numbers are described by parity alternating permutations.