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 language | English |
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Pages (from-to) | 29-57 |
Number of pages | 29 |
Journal | European Journal of Combinatorics |
Volume | 26 |
Issue number | 1 |
DOIs | |
State | Published - 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.
Funders | Funder number |
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European Commission | HPRN-CT-2001-00272 |