  Let $S_{g}$ denote the closed orientable surface of genus $g$. We construct
exponentially many mapping class group orbits of pairs of simple closed curves
which fill $S_{g}$ and intersect minimally, by showing that such orbits are in
correspondence with the solutions of a certain permutation equation in the
symmetric group. Next, we demonstrate that minimally intersecting filling pairs
are combinatorially optimal, in the sense that there are many simple closed
curves intersecting the pair exactly once. We conclude by initiating the study
of a topological Morse function $\mathcal{F}_{g}$ over the Moduli space of
Riemann surfaces of genus $g$, which, given a hyperbolic metric $\sigma$,
outputs the length of the shortest, minimally intersecting filling pair for the
metric $\sigma$. We completely characterize the global minima of
$\mathcal{F}_{g}$, and using the exponentially many mapping class group orbits
of minimally intersecting filling pairs that we construct in the first portion
of the paper, we show that the number of such minima grow at least
exponentially in $g$.
