On the number of dessins d’enfants with all permutations n-cycles

A formula is derived for the number of triples of [Formula: see text]-cycles in [Formula: see text], up to equivalence under simultaneous conjugation, whose product is [Formula: see text]. This collection of permutations coincides with the coverings of the Riemann sphere branched at three points and whose branch orders coincide with the multiplicity of the covering. While this formula has appeared as a special case in the work of Mednykh, the current proof illustrates a direct approach which avoids the use of character theory of the symmetric group. In particular, the proof focuses on developing a criterion for when an [Formula: see text]-cycle is fixed by permutations which are products of [Formula: see text] [Formula: see text]-cycles.