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 40 statistics 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 paths.
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.

6. Maps

We have the following 19 maps in the database:

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 binary word
bounce path
touch composition
decomposition reverse
rise composition
peeling map

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

DyckPaths (last edited 2015-12-02 16:32:57 by ChristianStump)