  We identify the dessin partition function with the partition function of the
Laguerre unitary ensemble (LUE). Combined with the result due to Cunden et al
on the relationship between the LUE correlators and strictly monotone Hurwitz
numbers introduced by Goulden et al, we then establish connection of dessin
counting to strictly monotone Hurwitz numbers. We also introduce a correction
factor for the dessin/LUE partition function, which plays an important role in
showing that the corrected dessin/LUE partition function is a tau-function of
the Toda lattice hierarchy. As an application, we use the approach of Dubrovin
and Zhang for the computation of the dessin correlators. In physicists'
terminology, we establish dualities among dessin counting, generalized Penner
model, and $\mathbb{P}^1$-topological sigma model.
