Dyck paths

1. Definition

A Dyck path of size $n$ is

Clearly equivalently, one can see a Dyck path as

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

3. Properties

4. Remarks

5. Statistics

We have the following 67 statistics on Dyck paths in the database:

The bounce statistic of a Dyck path.
The dinv statistic of a Dyck path.
The number of touch points of a Dyck path.
The area of a Dyck path.
The height of a Dyck path.
The number of parking functions supported by a Dyck path.
The number of peaks of a Dyck path.
The number of double rises of a Dyck path.
The number of initial rises of a Dyck path.
The position of the first return of a Dyck path.
The major index of a Dyck path.
The number of elements smaller than the given Dyck path in the Tamari Order.
The product of the heights of the descending steps of a Dyck path.
The number of valleys of a Dyck path not on the x-axis.
The number of valleys of the Dyck path.
The number of alternating sign matrices for a given Dyck path.
The number of centered tunnels of a Dyck path.
The number of left tunnels of a Dyck path.
The pyramid weight of the Dyck path.
The bounce count of a Dyck path.
The number of evenly positioned ascents, with the initial position equal to 1.
The number of upper interactions of a Dyck path.
The difference of lower and upper interactions.
The number of non-final maximal sub-paths of length greater than one.
The dinv deficit of a Dyck path.
The bounce deficit of a Dyck path.
The number of factors DDU in a Dyck path.
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
The sum of the heights of the peaks of a Dyck path.
The number of Dyck paths that are weakly below a Dyck path.
The number of Dyck paths that are weakly above a Dyck path, except for the path i....
The number of Dyck paths that are weakly above a Dyck path.
The number of Dyck paths that are weakly below a Dyck path, except for the path i....
The position of the last up step in a Dyck path.
The position of the first down step of a Dyck path.
The maximal area to the right of an up step of a Dyck path.
The number of long tunnels of a Dyck path.
The length of the maximal rise of a Dyck path.
The number of rises of length 1 of a Dyck path.
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
The number of global maxima of a Dyck path.
The sum of the areas of the rectangles formed by two consecutive peaks and the va....
The length of the minimal rise of a Dyck path.
The number of rises of length 2 of a Dyck path.
The number of rises of length at least 2 of a Dyck path.
The number of rises of length at least 3 of a Dyck path.
The number of rises of length 3 of a Dyck path.
The number of hills of a Dyck path.
The number of centered multitunnels of a Dyck path.
The number of odd rises of a Dyck path.
The number of up steps after the last double rise of a Dyck path.
The number of points below the Dyck path such that the diagonal to the north-east....
The global dimension of the LNakayama algebra associated to a Dyck path.
The dominant dimension of the LNakayama algebra associated to a Dyck path.
The finitistic dominant dimension of a Dyck path.
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path.
The global dimension minus the dominant dimension of the LNakayama algebra associ....
The maximal n such that the minimal generator-cogenerator module in the LNakayama....
The number of pairs of centered tunnels, one strictly containing the other, of a ....
The number of pairs of left tunnels, one strictly containing the other, of a Dyck....
The position of the last double rise in a Dyck path.
The logarithmic height of a Dyck path.
The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver.....
The number of occurrences of the pattern UUU in a Dyck path.
The number of occurrences of the pattern UDU in a Dyck path.
The sum of the skew hook positions in a Dyck path.
The major index of a Dyck path according to Zhao and Zhong.

6. Maps

We have the following 42 maps in the database:

to Dyck path
to Dyck path: up step, left tree, down step, right tree
to Tamari-corresponding Dyck path
to non-crossing permutation
to 321-avoiding permutation
to 132-avoiding permutation
to ordered tree
to partition
reverse
to Tamari-corresponding binary tree
zeta map
to 312-avoiding permutation
inverse zeta map
to two-row standard tableau
to binary tree: up step, left tree, down step, right tree
to alternating sign matrix
to Dyck path
to Dyck path
to Dyck path
to binary word
bounce path
touch composition
decomposition reverse
rise composition
peeling map
swap returns and last-descent
to 321-avoiding permutation (Krattenthaler)
Lalanne-Kreweras involution
Cori-Le Borgne involution
Elizalde-Deutsch bijection
Barnabei-Castronuovo involution
Adin-Bagno-Roichman transformation
left-to-right-maxima to Dyck path
to 321-avoiding permutation (Billey-Jockusch-Stanley)
switch returns and last double rise
to symmetric ASM
to noncrossing partition
logarithmic height to pruning number
pruning number to logarithmic height
promotion
inverse promotion
to noncrossing matching

7. References

[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.

8. Sage examples