Alternative proofs of the general dessin-counting proposition

Develop proofs other than the existing connected LUE-correlator-based proof for the more general Proposition A relating Grothendieck’s dessin counting to strictly monotone Hurwitz numbers, beyond the special case in which all parts of the partition are equal to l.

Background

The paper relates the weighted enumeration M^{[l]}_{g,k}(\nu_1,\ldots,\nu_k) of connected l-hypermaps to strictly monotone double Hurwitz numbers when the partition \mu consists entirely of parts equal to l. Combining this relation with a result from the literature yields an identity, Proposition A, expressing the weighted number N_{v,k}(\mu) of labelled Grothendieck dessins as a sum of strictly monotone Hurwitz numbers over partitions \nu of fixed degree and length.

For \mu=(l,\ldots,l), Proposition A specializes to equation (Nvk-l), which connects dessin counts with l-hypermap numbers. The paper notes that this specialized identity can be proved by purely combinatorial means, but explicitly states that alternative proofs of the more general Proposition A are not known.