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 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.
Highlights
Let c be the long cycle (1, 2, 3, . . . , n) in the symmetric group Sn
The number of minimal factorizations of the cycle c is nn−2, as was first shown by Denes [4]. One can interpret this result as the counting of the number of maximal chains in the lattice of noncrossing partitions of [1, n], whose definition is recalled in the text below
More generally one has: Lemma 4 Let π be a noncrossing partition and σπ = σπ z be a minimal factorization with a cycle z on k elements, π is obtained from π by splitting a block of π into an interval or near interval partition with k blocks
Summary
Let c be the long cycle (1, 2, 3, . . . , n) in the symmetric group Sn. It is elementary to see that at least n − 1 factors are needed to write c as a product of transpositions, such as c = (1, 2)(2, 3) . . . (n − 1, n). The number of minimal factorizations of the cycle c is nn−2, as was first shown by Denes [4] One can interpret this result as the counting of the number of maximal chains in the lattice of noncrossing partitions of [1, n], whose definition is recalled in the text below. One can interpret such factorizations as chains of a certain type in the lattice of noncrossing partitions and extend the definition of the weight wt to obtain the following. In the particular case of transposition factorizations (ai = 2 for all i), Theorem 1 is equivalent to a multivariate hook length formula for trees.
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