Geodesic
On decompositions of a cyclic permutation into a product of a given number of permutations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 1-6.

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The investigation of decompositions of a permutation into a product of permutations satisfying certain conditions plays a key role in the study of meromorphic functions or, equivalently, branched coverings of the 2-sphere; it goes back to A. Hurwitz' work in the late nineteenth century. In 2000 M. Bousquet-Melou and G. Schaeffer obtained an elegant formula for the number of decompositions of a permutation into a product of a given number of permutations corresponding to coverings of genus 0. Their formula has not been generalized to coverings of the sphere by surfaces of higher genera so far. This paper contains a new proof of the Bousquet-Melou–Schaeffer formula for the case of decompositions of a cyclic permutation, which, hopefully, can be generalized to positive genera.
Keywords: Hurwitz number
Mots-clés : Bousquet-Melou–Schaeffer formula. 
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B. S. Bychkov. On decompositions of a cyclic permutation into a product of a given number of permutations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 1-6. http://geodesic.mathdoc.fr/item/FAA_2015_49_2_a0/

[1] S. K. Lando, D. Zvonkin, “O kratnostyakh otobrazheniya Lyashko–Loiengi na stratakh diskriminanta”, Funkts. analiz i ego pril., 33:3 (1999), 21–34 | DOI | MR | Zbl

[2] M. Bousquet-Melou, G. Schaeffer, “Enumeration of planar constellations”, Adv. Appl. Math., 24:4 (2000), 337–368 | DOI | MR | Zbl

[3] T. Ekedahl, S. K. Lando, M. Shapiro, A. Vainshtein, “Hurwitz numbers and intersections on moduli spaces of curves”, Invent. Math., 146 (2001), 297–327 | DOI | MR | Zbl

[4] I. P. Goulden, D. M. Jackson, “The combinatorial relationship between trees, cacti and certain connection coefficients for the symmetric group”, European J. of Combin., 13:5 (1992), 357–365 | DOI | MR | Zbl

[5] I. P. Goulden, D. M. Jackson, “The KP-hierarchy, branched covers and triangulations”, Adv. Math., 219:3 (2008), 932–951 | DOI | MR | Zbl

[6] A. Hurwitz, “Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten”, Math. Ann., 39:1 (1891), 1–61 | DOI | MR

[7] P. Johnson, Double Hurwitz numbers via the infinite wedge, arXiv: 1008.3266 | MR

[8] E. Looijenga, “The complement of the bifurcation variety of a simple singularity”, Invent. Math., 23 (1974), 105–116 | DOI | MR | Zbl

[9] S. Shadrin, L. Spitz, D. Zvonkine, “On double Hurwitz numbers with completed cycles”, J. London Math. Soc., 86:2 (2012), 407–432 | DOI | MR | Zbl