Some identities for the Catalan and Fine numbers
Combinatorics
2007-05-23 v1
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 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.
Cite
@article{arxiv.math/0502532,
title = {Some identities for the Catalan and Fine numbers},
author = {David Callan},
journal= {arXiv preprint arXiv:math/0502532},
year = {2007}
}
Comments
LaTeX, 17 pages