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Results tagged with catalan-numbers
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user 47484
The Catalan numbers form the sequence of numbers starting 1,1,2,5,14,42,... with explicit formula $\frac{1}{n+1}\binom{2n}{n}$. It counts many combinatorial objects like planar binary trees, triangulations, noncrossing partitions, Dyck paths, etc. See https://oeis.org/A000108
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Looking for a combinatorial proof for a Catalan identity
By the ballot theorem, $\frac{k}{n} \binom{2n}{n+k}$ is the number of Dyck paths, i.e. $(1,1), (1,-1)$-walks in the quadrant, from the origin to $(2n-1, 2k-1)$. You need to concatenate a pair of those …
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