  It is well-known that the number of planar maps with prescribed vertex degree
distribution and suitable labeling can be represented as the leading
coefficient of the $\frac{1}{N}$-expansion of a joint cumulant of traces of
powers of an $N$-by-$N$ GUE matrix. Here we undertake the calculation of this
leading coefficient in a different way. Firstly, we tridiagonalize the GUE
matrix in the manner of Trotter and Dumitriu-Edelman and then alter it by
conjugation to make the subdiagonal identically equal to $1$. Secondly, we
apply the cluster expansion technique (specifically, the
Brydges-Kennedy-Abdesselam-Rivasseau formula) from rigorous statistical
mechanics. Thirdly, by sorting through the terms of the expansion thus
generated we arrive at an alternate interpretation for the leading coefficient
related to factorizations of the long cycle $(12\cdots n)\in S_n$. Finally, we
reconcile the group-theoretical objects emerging from our calculation with the
labeled mobiles of Bouttier-Di Francesco-Guitter.
