Prefixes of minimal factorisations of a cycle

TL;DR

This paper provides a new combinatorial proof for counting k-prefixes of minimal factorizations of an n-cycle into transpositions, using a surjection inspired by parking functions and Stanley's bijection.

Contribution

It introduces a surjective proof with constant fiber size for the enumeration of k-prefixes, extending previous bijective approaches in cycle factorization enumeration.

Findings

Counts k-prefixes of minimal factorizations as n^{k-1} * binomial(n, k+1)

Constructs a surjection with fibers of constant size for the enumeration

Connects minimal factorizations to parking functions through a surjective approach

Abstract

We give a bijective proof of the fact that the number of k-prefixes of minimal factorisations of the n-cycle (1...n) as a product of n-1 transpositions is n^{k-1}\binom{n}{k+1}. Rather than a bijection, we construct a surjection with fibres of constant size. This surjection is inspired by a bijection exhibited by Stanley between minimal factorisations of an n-cycle and parking functions, and by a counting argument for parking functions due to Pollak.