Some identities for the Catalan and Fine numbers

Details Some identities for the Catalan and Fine numbers Some identities for the Catalan and Fine numbers

2005-02-25

arxiv.org/abs/math/0502532v1

Mathematics

Combinatorics

LaTeX, 17 pages

Scientific paper

We establish combinatorial interpretations of several identities for the Catalan and Fine numbers and, along the way, we present some new bijections of independent interest. Briefly, we show that C_{n} = 1/(n+1) Sum_{k} (n+1)choose(2k+1) (n+k)choose(k) counts ordered trees on n edges by number of interior vertices adjacent to a leaf, and C_{n} = 2/(n+1) Sum_{k} (n+1)choose(k+2) (n-2)choose(k) counts Dyck n-paths by number of long interior inclines. We also give an analogue for the Fine numbers of Touchard's Catalan number identity.

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