Queries for Dyck paths: search statistic / browse statistics / browse maps from / browse maps to

# 1. 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*.

the 5 Dyck paths of size 3 | ||||

[1,0,1,0,1,0] |
[1,0,1,1,0,0] |
[1,1,0,0,1,0] |
[1,1,0,1,0,0] |
[1,1,1,0,0,0] |

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.

# 2. 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 $(2n,0)$.

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

# 3. Remarks

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.

# 4. References

*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.

# 5. Sage examples

# 6. Technical information for database usage

A Dyck path is uniquely represented as a

**Dyck word**.Dyck paths are

**graded by the semilength**.- The database contains all Dyck paths of semilength at most 8.