Some factorisations counted by Catalan numbers

Abstract

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 ( 1 2 … n + 1 ) into n transpositions is C n , the n -th Catalan number. A factorisation ( ( a 1 b 1 ) , ( a 2 b 2 ) , … , ( a n b n ) ) is primitive if its transpositions are “ordered”, in the sense that the a i s 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.