# Bijections for generalized Tamari intervals via orientations

## TL;DR

This paper introduces two new bijections connecting generalized Tamari intervals with non-separable maps, extending existing combinatorial correspondences and providing new generating function formulas.

## Contribution

It presents two novel bijections between generalized Tamari intervals and non-separable maps, expanding the combinatorial understanding and linking to existing structures.

## Key findings

- Two bijections between generalized Tamari intervals and non-separable maps.
- A trivariate generating function interpolating between Tamari and generalized Tamari intervals.
- Extension of the Bernardi-Bonichon bijection to new combinatorial objects.

## Abstract

Generalized Tamari intervals have been recently introduced by Pr\'eville-Ratelle and Viennot, and have been proved to be in bijection with (rooted planar) non-separable maps by Fang and Pr\'eville-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.