Annular noncrossing permutations and minimal transitive factorizations

TL;DR

This paper provides two combinatorial proofs for a formula counting minimal transitive factorizations of permutations with two cycles, utilizing recent results on annular noncrossing partitions of type B.

Contribution

It introduces two new combinatorial proofs for a known enumeration formula, connecting minimal transitive factorizations with annular noncrossing partitions.

Findings

Two combinatorial proofs of Goulden and Jackson's formula

Connection between minimal transitive factorizations and annular noncrossing partitions

Utilization of recent results on maximal chains of annular noncrossing partitions

Abstract

We give two combinatorial proofs of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type $B$.