Some identities for the Catalan and Fine numbers
Abstract
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 Cn = 1/(n+1) Sumk (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 Cn = 2/(n+1) Sumk (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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