Permutation statistics on the alternating group

Abstract

Let A n ⊆ S n denote the alternating and the symmetric groups on 1 , … , n . MacMahon's theorem [P.A. MacMahon, Combinatory Analysis I–II, Cambridge Univ. Press, 1916], about the equi-distribution of the length and the major indices in S n , has received far reaching refinements and generalizations, by Foata [Proc. Amer. Math. Soc. 19 (1968) 236], Carlitz [Trans. Amer. Math. Soc. 76 (1954) 332; Amer. Math. Monthly 82 (1975) 51], Foata-Schützenberger [Math. Nachr. 83 (1978) 143], Garsia–Gessel [Adv. Math. 31 (1979) 288] and followers. Our main goal is to find analogous statistics and identities for the alternating group A n . A new statistics for S n , the delent number, is introduced. This new statistics is involved with new S n identities, refining some of the results in [D. Foata, M.P. Schützenberger, Math. Nachr. 83 (1978) 143; A.M. Garsia, I. Gessel, Adv. Math. 31 (1979) 288]. By a certain covering map f : A n + 1 → S n , such S n identities are ‘lifted’ to A n + 1 , yielding the corresponding A n + 1 equi-distribution identities.