Chapter 4c:Trees and tableaux: alternative trees, non-ambiguous trees and beyond

Alternative trees were introduced by Nadeau in his study of alternative tableaux, tableaux introduced by the speaker in relation with the PASEP model in physics. We construct bijections between alternative tableaux, alternative trees and permutations. Non-ambiguous trees were introduced and studied by Aval, Boussicault, Bouvel, Silimbani, as a subclass of tree-like tableaux (which are in bijection with alternative tableaux). They show the relation with Bessel functions and gave a new hook formula for such trees. We give a new proof of this formula using the bijection between alternative tableaux and alternative trees. The relation with Bessel functions was explained by Emma Jin using the theory of heaps of pieces. I will finish this lecture by giving new bijections for binary trees (Catalan numbers) and for tree-like tableaux (enumerated by n!) inspired by the notion of "demultiplication of equations" in the quadratic algebra related to the PASEP model.