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Mesh patterns and the expansion of permutation statistics as sums of permutation patterns
Stockholm University, Faculty of Science, Department of Mathematics.
2011 (English)In: The Electronic Journal of Combinatorics, ISSN 1077-8926, Vol. 18, no 2, P5- p.Article in journal (Refereed) Published
Abstract [en]

Any permutation statistic f : G -> C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: f = Sigma(tau)lambda(f)(tau)tau . 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 = (pi, R) is an occurrence of the permutation pattern pi with additional restrictions specified by R on the relative position of the entries of the occurrence. We show that, for any mesh pattern p = (pi, R), wehave lambda(p)(tau) = (-1)(vertical bar tau vertical bar-vertical bar pi vertical bar)p*(tau) where p* = (pi, R(c)) 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, Andre permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

Place, publisher, year, edition, pages
2011. Vol. 18, no 2, P5- p.
National Category
URN: urn:nbn:se:su:diva-69429ISI: 000288371700002OAI: diva2:478266
authorCount :2Available from: 2012-01-16 Created: 2012-01-12 Last updated: 2012-01-16Bibliographically approved

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