On quasi-polynomials counting planar tight maps

Understand

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.