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A Simple Recurrence for Covers of the Sphere With Branch Points of Arbitrary Ramification
I.P. Goulden1 and Luis G. Serrano1, 1,2
1University of Waterloo, Department of Combinatorics & Optimization, Waterloo, Ontario N2L 3G1, Canada
ipgoulden@uwaterloo.ca
2University of Michigan, Department of Mathematics, Ann Arbor, MI 48109-1043, USA
lserrano@umich.edu
Annals of Combinatorics 10 (4) p.431-441 December, 2006
AMS Subject Classification: 05A15, 14H10, 58D29
Abstract:

The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity.

Goulden and Jackson have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the Join-Cut equation. Recently, Bousquet-Mélou and Schaeffer have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called m-Eulerian trees.

In this paper, we give a simple partial differential equation for Bousquet-Mélou and Schaeffer's generating series, and for Goulden and Jackson's generating series, as well as a new proof of the result by Bousquet-Mélou and Schaeffer. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-Mélou and Schaeffer's m-Eulerian trees.
Keywords: minimal, transitive, permutation factorizations, ramified covers, m-Eulerian trees, exact enumeration, generating functions

References:

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3. I.P. Goulden and D.M. Jackson, Combinatorial Enumeration, JohnWiley & Sons, New York, 1983.

4. I.P. Goulden and D.M. Jackson, Transitive factorizations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997) 51-60.

5. A. Hurwitz, Über Riemann'sche Flächen mit gegebenen Verzweignungspunkten, Math. Ann. 39 (1) (1891) 1-60.

6. A. Hurwitz, Über die Anzahl der Riemann'schen Flächen mit gegebenen Verzweignungspunkten, Math. Ann. 55 (1) (1901) 53-66.