Crossings and alignments of permutations

We introduce the notion of crossings and nestings of a permutation. We compute the generating function of permutations with a fixed number of weak exceedances, crossings and nestings. We link alignments and permutation patterns to these statistics. We generalize to the case of decorated permutations. Finally we show how this is related to the stationary distribution of the Partially ASymmetric Exclusion Process (PASEP) model.

[1]  Toufik Mansour,et al.  Counting occurrences of a pattern of type (1, 2) or (2, 1) in permutations , 2002, Adv. Appl. Math..

[2]  Doron Zeilberger,et al.  Denert's permutation statistic is indeed Euler-Mahonian , 1990 .

[3]  Sylvie Corteel,et al.  A Markov Chain on Permutations which Projects to the PASEP , 2006 .

[4]  Jiang Zeng,et al.  New Euler-Mahonian Statistics on Permutations and Words , 1997 .

[5]  Philippe Biane,et al.  Permutations Suivant le Type d'Exce'dance et le Nombre d'Inversions et Interpre'tation Combinatoire d'une Fraction Continue de Heine , 1993, Eur. J. Comb..

[6]  Philippe Flajolet,et al.  On congruences and continued fractions for some classical combinatorial quantities , 1982, Discret. Math..

[7]  Lauren K. Williams,et al.  Permutation tableaux and permutation patterns , 2007, J. Comb. Theory, Ser. A.

[8]  X. G. Viennot,et al.  Moments des q-Polynômes de Laguerre et la Bijection de Foata-Zeilberger , 1994 .

[9]  Rosena R. X. Du,et al.  Crossings and nestings of matchings and partitions , 2005, math/0501230.

[10]  B. Derrida,et al.  Exact solution of a 1d asymmetric exclusion model using a matrix formulation , 1993 .

[11]  Gérard Viennot,et al.  Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'euler et nombres de Genocchi , 1979, Discret. Math..

[12]  Martin Klazar.  On Identities Concerning the Numbers of Crossings and Nestings of Two Edges in Matchings , 2006, SIAM J. Discret. Math..

[13]  Sylvie Corteel,et al.  A Combinatorial Derivation of the PASEP Stationary State , 2006, Electron. J. Comb..

[14]  Jiang Zeng,et al.  Distribution of Crossings, Nestings and Alignments of Two Edges in Matchings and Partitions , 2006, Electron. J. Comb..

[15]  Gilles Schaeffer,et al.  A combinatorial approach to jumping particles , 2005, J. Comb. Theory, Ser. A.

[16]  Philippe Flajolet.  Combinatorial aspects of continued fractions , 1980, Discret. Math..

[17]  Philippe Biane,et al.  Permutations following excess type and number of inversions and combinatory interpretation of a continued fraction of Heine , 1993 .

[18]  Lauren K. Williams,et al.  Enumeration of totally positive Grassmann cells , 2003, math/0307271.

[19]  Sylvie Corteel,et al.  Tableaux combinatorics for the asymmetric exclusion process , 2007, Adv. Appl. Math..