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**Harvard**

Steingrimsson, E. och Williams, L. (2007) *Permutation tableaux and permutation patterns*.

** BibTeX **

@article{

Steingrimsson2007,

author={Steingrimsson, Einar and Williams, Lauren K.},

title={Permutation tableaux and permutation patterns},

journal={Journal of Combinatorial Theory Series A},

issn={0097-3165},

volume={114},

issue={2},

pages={211-234},

abstract={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 I-diagrams of Alex Postmkov [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. Efiksen, A. Reifegerste, and X. Viennot. (c) 2006 Elsevier Inc. All rights reserved.},

year={2007},

keywords={le-tableau, pen-nutation patterns, permutation tableaux, q-analogs, EULER, POLYNOMIALS },

}

** RefWorks **

RT Journal Article

SR Print

ID 81496

A1 Steingrimsson, Einar

A1 Williams, Lauren K.

T1 Permutation tableaux and permutation patterns

YR 2007

JF Journal of Combinatorial Theory Series A

SN 0097-3165

VO 114

IS 2

SP 211

OP 234

AB 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 I-diagrams of Alex Postmkov [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. Efiksen, A. Reifegerste, and X. Viennot. (c) 2006 Elsevier Inc. All rights reserved.

LA eng

DO 10.1016/j.jcta.2006.04.001

OL 30