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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, ₍ (d₁, , dₙ) \) of planar tight maps with \ (n\) labeled faces of prescribed degrees \ (d₁, , dₙ\), where a marked vertex is seen as a face of degree \ (0\). It is a quasi-polynomial in \ ( (d₁, , dₙ) \), 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
Bouttier et al. (Mon,) studied this question.