Annular noncrossing permutations and minimal transitive factorizations
  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 $B$. 
Jang Soo Kim, Seunghyun Seo, Heesung Shin
math.CO