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{\bf Sen-Peng Eu and Tung-Shan Fu}
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{\bf Exterior Pairs and Up Step Statistics on Dyck Paths}
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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 $h\equiv 0$ (mod
3)'. Moreover, for $m\ge 3$, we prove that the two statistics `number
of up steps at height $h$ with $h\equiv 0$ (mod $m$)' and `number of
up steps at height $h$ with $h\equiv m-1$ (mod $m$)' on the set
$\mathcal{C}_n$ are `almost equidistributed'. Both results are proved
combinatorially.
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