Some factorisations counted by Catalan numbers

In this paper, yet another occurrence of the Catalan numbers is presented; it is shown that the number of primitive factorisations of the cyclic permutation into n transpositions is Cn, the n-th Catalan number. A factorisation is primitive if its transpositions are “ordered”, in the sense that the ais are non-decreasing. We show that the sequence counting primitive factorisations satisfies the recurrence for Catalan numbers, and we exhibit an explicit bijection between the set of primitive factorisations and the set of 231-avoiding permutations, known to have size counted by Catalan numbers.