Explicit formulas for hypermaps and maps

Abstract

The study of hypermaps and maps is ubiquitous, as they are closely connected with geometry, mathematical physics, free probability and algebra. A universal 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.