Permutation Statistics on the Alternating Group

Let An ⊆ Sn denote the alternating and the symmetric groups on 1, . . . , n. MacMahaon’s theorem [11], about the equi-distribution of the length and the major indices in Sn, has received far reaching refinements and generalizations, by Foata [5], Carlitz [3, 4], FoataSchützenberger [6], Garsia-Gessel [7] and followers. Our main goal is to find analogous statistics and identities for the alternating group An. A new statistic for Sn, the delent number, is introduced. This new statistic is involved with new Sn equi-distribution identities, refining some of the results in [6] and [7]. By a certain covering map f : An+1 → Sn, such Sn identities are ‘lifted’ to An+1, yielding the corresponding An+1 equi-distribution identities. Partially supported by Minerva Grant No. 8441 and by EC’s IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272 Partially supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities and by EC’s IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272