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

Petter Brändén ; Anders Claesson

The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2

Résumé

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ 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 $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ 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, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

Détail

DOI : 10.37236/2001

Classification : 05A05, 05A15, 05A19
Mots-clés : permutation statistics, permutation pattern, mesh patterns, left-to-right maxima, descents, excedances, strong fixed points, major index, Andre permutations, simsun permutations, Mahonian statictics