Permutation statistics on the alternating group

Amitai Regev; Yuval Roichman

doi:10.1016/j.aam.2003.10.004

Permutation statistics on the alternating group

Amitai Regev
, Yuval Roichman

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

Let A_n⊆ S_ndenote 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
Pages (from-to)	676-709
Number of pages	34
Journal	Advances in Applied Mathematics
Volume	33
Issue number	4
DOIs	https://doi.org/10.1016/j.aam.2003.10.004
State	Published - Nov 2004

Bibliographical note

Funding Information:
* Corresponding author. E-mail addresses: [email protected] (A. Regev), [email protected] (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.

Funding

* Corresponding author. E-mail addresses: [email protected] (A. Regev), [email protected] (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.

Funders	Funder number
European Commission	HPRN-CT-2001-00272
Israel Academy of Sciences and Humanities
Israel Science Foundation

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10.1016/j.aam.2003.10.004

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title = "Permutation statistics on the alternating group",

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-Sch{\"u}tzenberger [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. Sch{\"u}tzenberger, 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.",

author = "Amitai Regev and Yuval Roichman",

note = "Funding Information: * Corresponding author. E-mail addresses: [email protected] (A. Regev), [email protected] (Y. Roichman). 1 Partially supported by Minerva Grant No. 8441 and by EC{\textquoteright}s IHRP Programme, within the Research Training Network {\textquotedblleft}Algebraic Combinatorics in Europe,{\textquotedblright} 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{\textquoteright}s IHRP Programme, within the Research Training Network {\textquotedblleft}Algebraic Combinatorics in Europe,{\textquotedblright} grant HPRN-CT-2001-00272.",

year = "2004",

month = nov,

doi = "10.1016/j.aam.2003.10.004",

language = "אנגלית",

volume = "33",

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AU - Regev, Amitai

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N1 - Funding Information: * Corresponding author. E-mail addresses: [email protected] (A. Regev), [email protected] (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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N2 - 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.

AB - 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.

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SN - 0196-8858

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JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

IS - 4

ER -

Permutation statistics on the alternating group

Abstract

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