Permutation statistics on the alternating group

Amitai Regev, Yuval Roichman

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13 Scopus citations

Abstract

Let An⊆ Sndenote 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 An. 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 : An+ 1 → Sn, such Sn identities are 'lifted' to An + 1, yielding the corresponding An + 1 equi-distribution identities.

Original languageEnglish
Pages (from-to)676-709
Number of pages34
JournalAdvances in Applied Mathematics
Volume33
Issue number4
DOIs
StatePublished - 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.

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