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Ribbon decomposition and twisted Hurwitz numbers
Yurii Burman1; Raphaël Fesler2
1 Higher School of Economics, Moscow, Russia, and Independent University of Moscow
2 Higher School of Economics, Moscow, Russia
Mathematics Research Reports, Volume 5 (2024), pp. 1-19.

Ribbon decomposition is a way to obtain a surface with boundary (compact, not necessarily oriented) from a collection of disks by joining them with narrow ribbons attached to segments of the boundary. Counting ribbon decompositions gives rise to a “twisted” version of the classical Hurwitz numbers (studied earlier by G. Chapuy and M. Dołęga [2] in a different context) and of the cut-and-join equation. We also provide an algebraic description of these numbers and an explicit formula for them in terms of zonal polynomials.

Received:
Revised:
Published online:
DOI: 10.5802/mrr.19
Classification: 57N05, 05C30
Keywords: surface with boundary, Hurwitz number, Jack polynomial
Yurii Burman 1; Raphaël Fesler 2

1 Higher School of Economics, Moscow, Russia, and Independent University of Moscow
2 Higher School of Economics, Moscow, Russia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Yurii Burman; Raphaël Fesler. Ribbon decomposition and twisted Hurwitz numbers. Mathematics Research Reports, Volume 5 (2024), pp. 1-19. doi : 10.5802/mrr.19. https://mrr.centre-mersenne.org/articles/10.5802/mrr.19/

[1] Yu. Burman; D. Zvonkine Cycle factorizations and 1-faced graph embeddings, European Journal of Combinatorics, Volume 31 (2010) no. 1, pp. 129-144 | DOI | MR | Zbl

[2] G. Chapuy; M. Dołęga Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions, Advances in Mathematics, Volume 409 (2022), 108645 | DOI | MR | Zbl

[3] A.F. Costa; S. Natanzon; B. Shapiro Topological classification of generic real meromorphic functions from compact surfaces, Annales Fennici Mathematici, Volume 43 (2018) no. 1, pp. 349-363 | MR | Zbl

[4] I. Goulden; A. Yong Tree-like properties of cycle factorizations, Journal of Combinatorial Theory Series A, Volume 98 (2002) no. 1, pp. 106-117 | DOI | MR | Zbl

[5] James E. Humphreys Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, 1992 | MR

[6] Maxim E. Kazarian; Sergey K. Lando An algebro-geometric proof of Witten’s conjecture, J. Amer. Math. Soc., Volume 20 (2007), pp. 1079-1089 | DOI | MR | Zbl

[7] M.E. Kazarian; S.K. Lando; S.M. Natanzon On framed simple purely real Hurwitz numbers, Izvestiya: Mathematics, Volume 85 (2021) no. 4, pp. 681-704 | DOI | Zbl

[8] Sergei K. Lando; Alexander K. Zvonkin Graphs on surfaces and their applications, EMS, 141, Springer, 2004 (with appendix by Don B. Zagier) | DOI

[9] Ian G. Macdonald Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, 29, The Clarendon Press, Oxford University Press, 1995 (with contributions by A. Zelevinsky) | DOI

[10] H. Markwig; J. Rau Tropical real Hurwitz numbers, Mathematische Zeitschrift, Volume 281 (2014) no. 1–2 | MR | Zbl

[11] S.M. Natanzon Simple Hurwitz numbers of a disk, Functional Analysis and Its Applications, Volume 44 (2010) no. 1, pp. 36-47 | DOI | Zbl

[12] S.M. Natanzon; B. Shapiro; A. Vainshtein Topological classification of generic real rational functions, J. Knot Theory Ramifications, Volume 11 (2002) no. 7, pp. 1063-1075 | DOI | MR | Zbl

[13] B. Shapiro; A. Vainshtein Counting real rational functions with all real critical values, Moscow Mathematical Journal, Volume 3 (2003) no. 2, pp. 647-659 | DOI | MR | Zbl

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