Bijections for generalized Tamari intervals via orientations

Abstract

We introduce two bijections for generalized Tamari intervals, which were recently introduced by Préville-Ratelle and Viennot, and proved to be in bijection with rooted non-separable maps by Fang and Préville-Ratelle. Our first construction proceeds via separating decompositions on quadrangulations and can be seen as an extension of the Bernardi-Bonichon bijection between Tamari intervals and minimal Schnyder woods. Our second construction directly exploits the Bernardi-Bonichon bijection and  the point of view of generalized Tamari intervals as a special case of classical Tamari intervals (synchronized Tamari intervals); it yields a trivariate generating function expression that interpolates between generalized Tamari intervals and classical Tamari intervals.