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Exterior pairs and up step statistics on Dyck paths. (English) Zbl 1217.05023

Summary: Let \(\mathcal{C}_n\) be the set of Dyck paths of length \(n\). In this paper, by a new automorphism of ordered trees, we prove that the statistic ‘number of exterior pairs’, introduced by A. Denise and R. Simion, on the set \(\mathcal{C}_n\) is equidistributed with the statistic ‘number of up steps at height \(h\) with \(h\equiv 0\pmod 3\)’. Moreover, for \(m\geq 3\), we prove that the two statistics ‘number of up steps at height \(h\) with \(h\equiv 0 \pmod m\)’ and ‘number of up steps at height \(h\) with \(h\equiv m-1 \pmod m\)’ on the set \(\mathcal{C}_n\) are ‘almost equidistributed’. Both results are proved combinatorially.

MSC:

05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
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