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Minimal factorizations of a cycle: a multivariate generating function
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020) Cet article a éte moissonné depuis la source Episciences

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It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions. 
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     author = {Biane, Philippe and Josuat-Verg\`es, Matthieu},
     title = {Minimal factorizations of a cycle: a multivariate generating function},
     journal = {Discrete mathematics & theoretical computer science},
     year = {2020},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     doi = {10.46298/dmtcs.6318},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6318/}
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Biane, Philippe; Josuat-Vergès, Matthieu. Minimal factorizations of a cycle: a multivariate generating function. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi: 10.46298/dmtcs.6318

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