The structure of alternative tableaux

In this paper we study alternative tableaux introduced by Viennot [X. Viennot, Alternative tableaux, permutations and partially asymmetric exclusion process, talk in Cambridge, 2008]. These tableaux are in simple bijection with permutation tableaux, defined previously by Postnikov [A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1 [math.CO], 2006]. We exhibit a simple recursive structure for alternative tableaux, from which we can easily deduce a number of enumerative results. We also give bijections between these tableaux and certain classes of labeled trees. Finally, we exhibit a bijection with permutations, and relate it to some other bijections that already appeared in the literature.

[1]  Thomas Lam,et al.  Total positivity for cominuscule Grassmannians , 2007 .

[2]  Gilbert Labelle,et al.  Combinatorial species and tree-like structures , 1997, Encyclopedia of mathematics and its applications.

[3]  Sylvie Corteel,et al.  Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials , 2009, Proceedings of the National Academy of Sciences.

[4]  Sylvie Corteel,et al.  Bijections for permutation tableaux , 2009, Eur. J. Comb..

[5]  M. Wadati,et al.  Asymmetric simple exclusion process with open boundaries and Askey–Wilson polynomials , 2003, cond-mat/0312457.

[6]  Alexander Burstein.  On Some Properties of Permutation Tableaux , 2007 .

[7]  Sylvie Corteel,et al.  Tableaux Combinatorics for the Asymmetric Exclusion Process II , 2008 .

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

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

[10]  Alexander Postnikov,et al.  Total positivity, Grassmannians, and networks , 2006 .

[11]  Sylvie Corteel,et al.  Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials , 2009 .

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

[13]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[14]  Xavier G'erard Viennot.  Canopy of binary trees, Catalan tableaux and the asymmetric exclusion process , 2009 .

[15]  Sylvie Corteel,et al.  Permutation tableaux and the asymmetric exclusion process , 2006 .

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

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

[18]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .