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.