Picture of virus under microscope

Research under the microscope...

The Strathprints institutional repository is a digital archive of University of Strathclyde research outputs.

Strathprints serves world leading Open Access research by the University of Strathclyde, including research by the Strathclyde Institute of Pharmacy and Biomedical Sciences (SIPBS), where research centres such as the Industrial Biotechnology Innovation Centre (IBioIC), the Cancer Research UK Formulation Unit, SeaBioTech and the Centre for Biophotonics are based.

Explore SIPBS research

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

Brändén, Petter and Claesson, Anders (2011) Mesh patterns and the expansion of permutation statistics as sums of permutation patterns. The Electronic Journal of Combinatorics, 18 (2).

Full text not available in this repository. (Request a copy from the Strathclyde author)

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.