On quasi-polynomials counting planar tight maps

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 \(N0,n(d_1,…,d_n)\) 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.Mathematics Subject Classifications: 05A15, 05A19Keywords: Planar maps, bijective enumeration, slice decomposition