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.

Mathematics Subject Classification ID

Exact enumeration problems, generating functions (05A15) Combinatorial aspects of representation theory (05E10)

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