Definition & Example

A Dyck path of semilength $n$ is a lattice path from $(0,0)$ to $(2n,0)$ consisting of $n$ up steps of the form $(1,1)$ and $n$ down steps of the form $(-1,1)$ which never goes below the x-axis $y=0$. Denote all Dyck paths of size $n$ by $\mathfrak{D}_n$.

Equivalently, a Dyck path of semilength $n$ can be seen as

a lattice path from $(0,0)$ to $(n,n)$ consisting of $n$ up steps of the form $(1,0)$ and $n$ down steps of the form $(0,1)$ which never goes below the diagonal $y=x$.
a Dyck word, this is a word in $\{0,1\}$ such that any prefix contains at least as many $1$'s as it contains $0$'s. Seeing a $1$ as an opening bracket and a $0$ as a closing bracket, Dyck words can be seen as well-formed bracketing systems.

There are $\operatorname{Cat}(n) = \frac{1}{n+1}\binom{2n}{n}$ Dyck paths of semilength $n$, see OEIS:A000108. This can for example be seen using the reflection principle.

Properties

A Dyck path $D$ of size $n+1$ can be decomposed into a Dyck path $D_1$ of size $k$ and a Dyck path $D_2$ of size $n-k$, where
$(2k+2,0)$ is the first touch point of $D$ at the x-axis,
$D_1$ is the prefix of $D$ from $(1,1)$ to $(2k-1,1)$ never going below the line $y=1$, and where
$D_2$ is the suffix of $D$ from $(2k,0)$ to $(2n,0)$.

This yields the recurrence

$$\operatorname{Cat}(n+1) = \sum_{k=1}^n \operatorname{Cat}(k) \cdot \operatorname{Cat}(n-k).$$

There is a natural extension of Dyck paths to $m$-Dyck paths which can be seen as lattice paths from $(0,0)$ to $(mn,n)$ that never go below the diagonal $y = \frac{1}{m} x$. The number of $m$-Dyck paths is given by the Fuss-Catalan number $C^{(m)}(n) = \frac{1}{mn+1} \binom{(m+1)n}{n}$ [Kra89].
Dyck paths and $m$-Dyck paths can be seen as type A instances of a more general combinatorial object for root systems, namely positive chambers in the (generalized) Shi arrangement [Ath04].
Dyck paths of semilength $n$ are in canonical bijection with Nakayama algebras with linear quiver and $n+1$ simple modules.

[Ath04] C.A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc., 36 (2004), pp. 294-392
[Kra89] C. Krattenthaler, Counting lattice paths with a linear boundary II, Sitz.ber. d. ÖAW Math.-naturwiss. Klasse 198 (1989), 171-199

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