Mesh patterns and the expansion of permutation statistics as sums of permutation patterns

Petter Brändén, Anders Claesson

Research output: Contribution to journalArticle

32 Citations (Scopus)

Abstract

Any permutation statistic ƒ : S → C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: ƒ= ∑rλƒ (τ ) τ. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern p = (π,R) is an occurrence of the permutation pattern with additional restrictions specified by R on the relative position of the entries of the occurrence. We show that, for any mesh pattern p = (π,R), we have λp(τ ) = (−1)|τ|−|π|p⋆( ) where p⋆ = (π,Rc) is the mesh pattern with the same underlying permutation as p but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.
LanguageEnglish
Article numberP5
Number of pages14
JournalThe Electronic Journal of Combinatorics
Volume18
Issue number2
Publication statusPublished - 13 Mar 2011

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Permutation Statistics
Permutation
Statistics
Mesh
Fixed point
Major Index
Restriction
Descent
Expand
Linear Combination

Keywords

  • permutation statistics
  • mesh patterns
  • André permutations
  • simsun permutations
  • Mahonian statistics

Cite this

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Mesh patterns and the expansion of permutation statistics as sums of permutation patterns. / Brändén, Petter; Claesson, Anders.

In: The Electronic Journal of Combinatorics, Vol. 18, No. 2, P5, 13.03.2011.

Research output: Contribution to journalArticle

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AU - Claesson, Anders

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AB - Any permutation statistic ƒ : S → C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: ƒ= ∑rλƒ (τ ) τ. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern p = (π,R) is an occurrence of the permutation pattern with additional restrictions specified by R on the relative position of the entries of the occurrence. We show that, for any mesh pattern p = (π,R), we have λp(τ ) = (−1)|τ|−|π|p⋆( ) where p⋆ = (π,Rc) is the mesh pattern with the same underlying permutation as p but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

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