On quasi-polynomials counting planar tight maps

TL;DR

This paper derives an explicit, bijective formula for counting planar tight maps with prescribed face degrees, revealing their quasi-polynomial nature and extending classical map enumeration results.

Contribution

It provides a new explicit formula for counting planar tight maps, generalizes Tutte's slicing formula to non-bipartite maps, and introduces a bijective derivation based on slice decomposition.

Findings

Number of planar tight maps is a quasi-polynomial in face degrees.

Derived an explicit bijective formula for counting tight maps.

Extended Tutte's slicing formula to non-bipartite maps.

Abstract

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 $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees $d_1,\ldots,d_n$, where a marked vertex is seen as a face of degree $0$. It is a quasi-polynomial in $(d_1,\ldots,d_n)$, 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.