Abstract
The aim of this paper is to find explicit formulae for the number of rooted hypermaps with a given number of darts on an orientable surface of genus \(g\le 3\). Such formulae were obtained earlier for \(g=0\) and \(g=1\) by Walsh and Arques respectively. We first employ the Egorychev’s method of counting combinatorial sums to obtain a new version of the Arques formula for genus \(g=1\). Then we apply the same approach to get new results for genus \(g=2,3\). We could do it due to recent results by Giorgetti, Walsh, and Kazarian, Zograf who derived two different, but equivalent, forms of the generating functions for the number of hypermaps of genus two and three.
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