On quasi-polynomials counting planar tight maps

Jérémie Bouttier (1) , Emmanuel Guitter (1) , Grégory Miermont (2)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

A tight map is a map with some of its vertices marked, such that every vertex of degree 1 is marked. We give an explicit formula for the number $N_{0,n}(d_1,…,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees $d_1,…,d_n$, where a marked vertex is seen as a face of degree 0. It is a quasi-polynomial in $(d_1,…,d_n)$, as shown previously by Norbury. Our derivation is bijective and based on the slice decomposition of planar maps. In the non-bipartite case, we also rely on enumeration results for two-type forests. We discuss the connection with the enumeration of non necessarily tight maps. In particular, we provide a generalization of Tutte's classical slicings formula to all non-bipartite maps.

Domaines: Combinatoire [math.CO]

Jérémie Bouttier, Emmanuel Guitter, Grégory Miermont. On quasi-polynomials counting planar tight maps. 2022. ⟨hal-03622398⟩