More bijective Catalan combinatorics on permutations and on signed permutations

In this paper, we construct bijections between Dyck paths, noncrossing partitions, and 231-avoiding permutations, which send the area statistic on Dyck paths to the inversion number on noncrossing partitions and on 231-avoiding permutations. This bijection has the additional property that it simultaneously sends the major index on Dyck paths to the sum of the major index and the inverse major index on noncrossing partitions and on 231-avoiding permutations, respectively. Moreover, we provide generalizations of these constructions to the group of signed permutations.

Keywords

bijective combinatorics, Catalan combinatorics, Dyck paths, noncrossing partitions, pattern-avoiding permutations, permutation statistics, major index

2010 Mathematics Subject Classification

Primary 05A19. Secondary 05A05, 05A18, 20F55.

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Published 27 December 2013