Generalized statistics on Sn and pattern avoidance

Amitai Regev, Yuval Roichman

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Natural q analogues of classical statistics on the symmetric groups Sn are introduced; parameters like: the q-length, the q-inversion number, the q-descent number and the q-major index. Here q is a positive integer. MacMahon's theorem (Combinatory Analysis I-II (1916)) about the equi-distribution of the inversion number and the reverse major index is generalized to all positive integers q. It is also shown that the q-inversion number and the q-reverse major index are equi-distributed over subsets of permutations avoiding certain patterns. Natural q analogues of the Bell and the Stirling numbers are related to these q statistics-through the counting of the above pattern-avoiding permutations.

Original languageEnglish
Pages (from-to)29-57
Number of pages29
JournalEuropean Journal of Combinatorics
Volume26
Issue number1
DOIs
StatePublished - Jan 2005

Bibliographical note

Funding Information:
The authors would like to thank Dominique Foata for some helpful remarks. A. Regev was 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. Y. Roichman was partially supported by EC’s IHRP Programme, within the Research Training Network ‘Algebraic Combinatorics in Europe’, grant HPRN-CT-2001-00272.

Funding

The authors would like to thank Dominique Foata for some helpful remarks. A. Regev was 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. Y. Roichman was partially supported by EC’s IHRP Programme, within the Research Training Network ‘Algebraic Combinatorics in Europe’, grant HPRN-CT-2001-00272.

FundersFunder number
European CommissionHPRN-CT-2001-00272

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