European Journal of Combinatorics
Specializations of generalized Laguerre polynomials
SIAM Journal on Mathematical Analysis - Special issue: the articles in this issue are dedicated to Richard Askey and Frank Olver
Regular Article: Moments des q-Polyn么mes de Laguerre et la Bijection de Foata-Zeilberger
Advances in Applied Mathematics
Regular Article: New Euler驴Mahonian Statistics on Permutations and Words
Advances in Applied Mathematics
Combinatorial Enumeration
Actions on permutations and unimodality of descent polynomials
European Journal of Combinatorics
Bijections for permutation tableaux
European Journal of Combinatorics
Permutations with extremal number of fixed points
Journal of Combinatorial Theory Series A
The q-tangent and q-secant numbers via continued fractions
European Journal of Combinatorics
A q-enumeration of alternating permutations
European Journal of Combinatorics
Bijections between pattern-avoiding fillings of Young diagrams
Journal of Combinatorial Theory Series A
New interpretations for noncrossing partitions of classical types
Journal of Combinatorial Theory Series A
The structure of alternative tableaux
Journal of Combinatorial Theory Series A
Combinatorics on permutation tableaux of type A and type B
European Journal of Combinatorics
Theoretical Computer Science
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In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the -diagrams of Alex Postnikov [A. Postnikov, Webs in totally positive Grassmann cells, in preparation; L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative sizes of pairs of letters in a permutation and their places. We call these statistics weak excedance statistics because of their close relation to weak excedances. The second bijection carries tableaux statistics (via the weak excedance statistics) to statistics based on generalized permutation patterns. We then give enumerative applications of these bijections. One nice consequence of these results is that the polynomial enumerating permutation tableaux according to their content generalizes both Carlitz'q-analog of the Eulerian numbers [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350] and the more recent q-analog of the Eulerian numbers found in [L. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319-342]. We conclude our paper with a list of open problems, as well as remarks on progress on these problems which has been made by A. Burstein, S. Corteel, N. Eriksen, A. Reifegerste, and X. Viennot.