Picture of a black hole

Strathclyde Open Access research that creates ripples...

The Strathprints institutional repository is a digital archive of University of Strathclyde's Open Access research outputs. Strathprints provides access to thousands of research papers by University of Strathclyde researchers, including by Strathclyde physicists involved in observing gravitational waves and black hole mergers as part of the Laser Interferometer Gravitational-Wave Observatory (LIGO) - but also other internationally significant research from the Department of Physics. Discover why Strathclyde's physics research is making ripples...

Strathprints also exposes world leading research from the Faculties of Science, Engineering, Humanities & Social Sciences, and from the Strathclyde Business School.

Discover more...

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)


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.