Connected permutations and hypermaps

In this lecture I will introduce the notion of hypermap~: a pair of permutations representing the embedding of graphs on surfaces. A nice bijection, due to Ossona de Mendez and Rosenstiehl, showing a coding of hypermaps by connected permutations, will be presented. This will be used to give the structure of a family of polynomials in two variables defined by weighted Dyck paths and whose coefficients are related to Stirling numbers.