Dyck paths

# 1. Definition

A **Dyck path** of size $n$ 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 $y=0$.

Clearly equivalently, one can see a Dyck path 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 path can also be identified with its **Dyck word** being $(0,1)$-sequence with $1$'s representing up steps and $0$'s representing down steps. Denote all Dyck paths of size $n$ by $\mathfrak{D}_n$.

# 2. Examples

$\mathfrak{D}_1 = \{ [1,0] \}$

$\mathfrak{D}_2 = \{ [1,0,1,0],[1,1,0,0] \}$

$\mathfrak{D}_3 = \{[1,0,1,0,1,0], [1,1,0,0,1,0], [1,1,1,0,0,0], [1,0,1,1,0,0], [1,1,0,1,0,0] \}$

$\mathfrak{D}_4 = $

# 3. Properties

The cardinality of $\mathfrak{D}_n$ is the $n^{th}$ Catalan number $\operatorname{Cat}_n = \frac{1}{n+1} \binom{2n}{n}$. This can for example be seen using the reflection principle.

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 $\operatorname{Cat}_{n+1} = \sum_{k=1}^n \operatorname{Cat}_k \operatorname{Cat}_{n-k}$.

# 4. 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**$C_n^{(m)} = \frac{1}{mn+1} \binom{(m+1)n}{n}$ [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].

# 5. Statistics

We have the following **19 statistics** in the database:

# 6. Maps

The

**reverse**of a Dyck path is the Dyck path obtained by horizontally flipping to (i.e., such that the points $(i,j)$ and $(2n-i,j)$ are interchanged).Obviously, this operation of Dyck paths of size $n$ is an involution.

In terms of Dyck words, this corresponds to sending a $(0,1)$-sequence $[d_1,d_2,\ldots,d_{2n}]$ to the sequence $[1-d_{2n},1-d_{2n-1},\ldots,1-d_1]$.

The

**partition**$\lambda(D) = (\lambda_1,\ldots,\lambda_{n-1})$ associated to a Dyck path $D$ is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.In terms of Dyck words, $\lambda_{i}$ is the number of $0$'s before the $(n+1-i)$-th $1$.

This map describes a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.

We have $\operatorname{area}(D) + |\lambda(D)| = \binom{n}{2}$.

The

**touch composition**is the composition corresponding to the set $\{ i-1 : a_i=0\hbox{ for }2 \leq i \leq n\}$, the**bounce composition**would be the composition corresponding to the touch points of the bounce path.

The following maps have individual pages with further explanations:

(area,dinv) to (bounce,area) map [LW02, Theorem 1]

# 7. References

*Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes*, Bull. London Math. Soc.,

**36**(2004), pp. 294-392.

[Deu99b] E. Deutsch, *An involution on Dyck paths and its consequences*, Discrete Math. **204** (1999), pp. 163-166.

[Kra89] C. Krattenthaler, *Counting lattice paths with a linear boundary II*, Sitz.ber. d. ÖAW Math.-naturwiss. Klasse **198** (1989), 171-199.

[LW02] N. Loehr, G. Warrington, *Square q, t-lattice paths and $\nabla(p_n)$*, Transactions of the AMS **359**(2) (2007).

# 8. Sage examples