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Fusy, Éric; Humbert, Abel
European journal of combinatorics, 03/2024, Letnik: 117Journal Article
Generalized Tamari intervals have been recently introduced by Préville-Ratelle and Viennot, and have been proved to be in bijection with (rooted planar) non-separable maps by Fang and Préville-Ratelle. We present two new bijections between generalized Tamari intervals and non-separable maps. Our first construction proceeds via separating decompositions on simple bipartite quadrangulations (which are known to be in bijection with non-separable maps). It can be seen as an extension of the Bernardi–Bonichon bijection between Tamari intervals and minimal Schnyder woods. On the other hand, our second construction relies on a specialization of the Bernardi–Bonichon bijection to so-called synchronized Tamari intervals, which are known to be in one-to-one correspondence with generalized Tamari intervals. It yields a trivariate generating function expression that interpolates between the bivariate generating function for generalized Tamari intervals, and the univariate generating function for Tamari intervals.
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