The Catalan number $C_n:=\mid {\mathcal{D}}_n\mid$

For $n\in \mathbb{N}$ the Catalan number $C_n$ is the number of Dyck paths of semilength $n$:

$C_n:=\mid {\mathcal{D}}_n\mid \in \mathbb{N}$

(Dyck paths of semilength $n$, The cardinality $\mid A\mid$ of a finite set). This is a natural number because ${\mathcal{D}}_n$ is finite (${\mathcal{D}}_n$ is a finite set), and the cardinality notation is defined for finite sets only.

$C_0=1$ and $C_1=1$, both read off the definition rather than stipulated: ${\mathcal{D}}_0$ is the one-element set containing the empty path, and ${\mathcal{D}}_1$ is the one-element set whose member has step word $UD$ (Dyck paths of semilength $n$).

$C_n\ge 1$ for every $n$, since ${\mathcal{D}}_n$ is nonempty (${\mathcal{D}}_n$ is a finite set).

• The Catalan number is defined as a count, and every formula for it is a theorem. Defining $C_n$ by a closed expression would make $C_0=1$ a convention about an empty product or an empty binomial coefficient, and would make the statement that the expression is a natural number something to be arranged rather than proved. Here integrality is free and the closed formula has content.

• The indexing convention. $C_n$ counts the Dyck paths of semilength $n$, equivalently the ballot words of length $2n$, so $C_0=1$ and the path has $2n$ steps. This is the indexing of Krattenthaler §10.3 and of Guichard §3.5, and every source consulted for this page agrees on it; a source indexing by the number of steps would call the same number $C_{2n}$, and no statement here is stated that way.