## 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, he 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.

Original language | English |
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Pages (from-to) | 676-709 |

Number of pages | 34 |

Journal | Advances in Applied Mathematics |

Volume | 33 |

Issue number | 4 |

DOIs | |

State | Published - Nov 2004 |

### Bibliographical note

Funding Information:* Corresponding author. E-mail addresses: regev@wisdom.weizmann.ac.il (A. Regev), yuvalr@math.biu.ac.il (Y. Roichman). 1 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. 2 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.