Geodesic
Bijective Enumeration of Bicolored Maps of Given Vertex Degree Distribution
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009).

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We derive a new formula for the number of factorizations of a full cycle into an ordered product of two permutations of given cycle types. For the first time, a purely combinatorial argument involving a bijective description of bicolored maps of specified vertex degree distribution is used. All the previous results in the field rely either partially or totally on a character theoretic approach. The combinatorial proof relies on a new bijection extending the one in [G. Schaeffer and E. Vassilieva. $\textit{J. Comb. Theory Ser. A}$, 115(6):903―924, 2008] that focused only on the number of cycles. As a salient ingredient, we introduce the notion of thorn trees of given vertex degree distribution which are recursive planar objects allowing simple description of maps of arbitrary genus. \par 
@article{DMTCS_2009_special_256_a4,
     author = {Morales, Alejandro and Vassilieva, Ekaterina},
     title = {Bijective {Enumeration} of {Bicolored} {Maps} of {Given} {Vertex} {Degree} {Distribution}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)},
     year = {2009},
     doi = {10.46298/dmtcs.2682},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2682/}
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%A Vassilieva, Ekaterina
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Morales, Alejandro; Vassilieva, Ekaterina. Bijective Enumeration of Bicolored Maps of Given Vertex Degree Distribution. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009). doi : 10.46298/dmtcs.2682. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2682/

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