  The study of hypermaps and maps is ubiquitous, as they are closely connected
with geometry, mathematical physics, free probability and algebra. A novel
recursion generalizing a fundamental identity of Frobenius which enumerates
factorizations of a permutation in group algebra theory has been discovered by
the author recently. Here we apply the recursion to study hypermaps as well as
maps and obtain a plethora of results in a unified way. For instance, we
succinctly provide a short proof of the celebrated Harer-Zagier formula and
present a general explicit formula for one-face hypermaps. As special cases of
the latter, we obtain simple explicit formulas for the numbers of ways of
expressing a long cycle as a product of a permutation of cycle-type $[1^p,
n-p]$ and, respectively, $[p, n-p]$ and a permutation with $m$ cycles for any
$p$ and $m$. To the best of our knowledge, only the cases for $p=0,\, 1$ are
known before.
