Some factorisations counted by Catalan numbers

Authors:
Daniele A. Gewurz;Francesca Merola
Affiliations:
Dipartimento di Matematica, Università di Roma "La Sapienza", Roma, Italy;Dipartimento di Matematica, Università di Roma "La Sapienza", Roma, Italy
Venue:
European Journal of Combinatorics
Year:
2006

Citing 4
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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 Cn, the n-th Catalan number. A factorisation ((a1 b1), (a2 b2),..., (an bn)) 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.