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arXiv · 1610.02982

Minimal factorizations of a cycle: a multivariate generating function

Philippe Biane·Matthieu Josuat-Vergès
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Abstract

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 $n^{k-1}$ where $n$ is the rank of the underlying symmetric group and $k$ is the number of factors. In particular, this is $n^{n-2}$ 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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Philippe Biane, Matthieu Josuat-Vergès. 2016-10-10. Minimal factorizations of a cycle: a multivariate generating function. https://arxiv.org/abs/1610.02982

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