Abstract
We prove in two different ways that the number of distinct prefixes of length $k$ of minimal factorisations of the $n$-cycle $(1\ldots n)$ as a product of $n-1$ transpositions is $\binom{n}{k+1}n^{k-1}$. Our first proof is not bijective but makes use of a correspondence between minimal factorisations and Cayley trees. The second proof consists of establishing a bijection between the set which we want to enumerate and the set of parking functions of a certain kind, which can be counted by a standard conjugation argument.
Highlights
It is very well known that the n-cycle (1 2 . . . n) cannot be written as a product of less than n − 1 transpositions and that there are nn−2 distinct ways of writing it as a product of exactly n − 1 transpositions
The second proof consists in establishing a bijection between the set which we want to enumerate and the set of parking functions of a certain kind, which can be counted by a standard conjugation argument
The sequencen 1 counts a variety of other combinatorial objects, including Cayley trees and parking functions, and a wealth of bijections have been described between minimal factorisations, Cayley trees and parking functions
Summary
It is very well known that the n-cycle (1 2 . . . n) cannot be written as a product of less than n − 1 transpositions and that there are nn−2 distinct ways of writing it as a product of exactly n − 1 transpositions. In the course of the study of the distribution of the eigenvalues of certain random unitary matrices, using the relations between the unitary groups and the symmetric groups (see [12]), we were led to enumerating the sequences of transpositions which appear as initial segments of a minimal factorisation of the n-cycle
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