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Catalan Numbers, Their Generalization, and Their Uses

The Mathematical Intelligencer, 1991
The authors are concerned with four enumeration problems all involving the numbers \(f(p,n)=(pn)!/n!(pn-n+1)!\) for positive integers \(p-1\) and \(n\). (1) Let \(a(p,n)\) be the number of planted plane trees having exactly \(n\) vertices with out-degree \(n\).
Hilton, Peter, Pedersen, Jean
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Catalan Numbers

2015
Catalan numbers are probably the most ubiquitous sequence of numbers in mathematics. This book gives for the first time a comprehensive collection of their properties and applications to combinatorics, algebra, analysis, number theory, probability theory, geometry, topology, and other areas.
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The Catalan Numbers

2015
The most important special case of Dyck words comes when \( a=b \) so that the words have the same number of dominant and nondominant letters. A slight variation on these Dyck words is ballot , which we also define here for completeness.
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Further identities on Catalan numbers

Discrete Mathematics, 2018
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Catalan Numbers