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Identifier

St001800: Dyck paths ⟶ ℤ

Values

[] => 1

[1,0] => 1

[1,0,1,0] => 2

[1,1,0,0] => 1

[1,0,1,0,1,0] => 4

[1,0,1,1,0,0] => 3

[1,1,0,0,1,0] => 3

[1,1,0,1,0,0] => 1

[1,1,1,0,0,0] => 1

[1,0,1,0,1,0,1,0] => 8

[1,0,1,0,1,1,0,0] => 6

[1,0,1,1,0,0,1,0] => 9

[1,0,1,1,0,1,0,0] => 3

[1,0,1,1,1,0,0,0] => 4

[1,1,0,0,1,0,1,0] => 6

[1,1,0,0,1,1,0,0] => 6

[1,1,0,1,0,0,1,0] => 3

[1,1,0,1,0,1,0,0] => 2

[1,1,0,1,1,0,0,0] => 1

[1,1,1,0,0,0,1,0] => 4

[1,1,1,0,0,1,0,0] => 1

[1,1,1,0,1,0,0,0] => 1

[1,1,1,1,0,0,0,0] => 1

[1,0,1,0,1,0,1,0,1,0] => 16

[1,0,1,0,1,0,1,1,0,0] => 12

[1,0,1,0,1,1,0,0,1,0] => 18

[1,0,1,0,1,1,0,1,0,0] => 6

[1,0,1,0,1,1,1,0,0,0] => 8

[1,0,1,1,0,0,1,0,1,0] => 18

[1,0,1,1,0,0,1,1,0,0] => 18

[1,0,1,1,0,1,0,0,1,0] => 9

[1,0,1,1,0,1,0,1,0,0] => 6

[1,0,1,1,0,1,1,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0] => 16

[1,0,1,1,1,0,0,1,0,0] => 4

[1,0,1,1,1,0,1,0,0,0] => 4

[1,0,1,1,1,1,0,0,0,0] => 5

[1,1,0,0,1,0,1,0,1,0] => 12

[1,1,0,0,1,0,1,1,0,0] => 9

[1,1,0,0,1,1,0,0,1,0] => 18

[1,1,0,0,1,1,0,1,0,0] => 6

[1,1,0,0,1,1,1,0,0,0] => 10

[1,1,0,1,0,0,1,0,1,0] => 6

[1,1,0,1,0,0,1,1,0,0] => 6

[1,1,0,1,0,1,0,0,1,0] => 6

[1,1,0,1,0,1,0,1,0,0] => 4

[1,1,0,1,0,1,1,0,0,0] => 3

[1,1,0,1,1,0,0,0,1,0] => 4

[1,1,0,1,1,0,0,1,0,0] => 1

[1,1,0,1,1,0,1,0,0,0] => 2

[1,1,0,1,1,1,0,0,0,0] => 1

[1,1,1,0,0,0,1,0,1,0] => 8

[1,1,1,0,0,0,1,1,0,0] => 10

[1,1,1,0,0,1,0,0,1,0] => 3

[1,1,1,0,0,1,0,1,0,0] => 3

[1,1,1,0,0,1,1,0,0,0] => 1

[1,1,1,0,1,0,0,0,1,0] => 4

[1,1,1,0,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,1,0,0,0] => 1

[1,1,1,0,1,1,0,0,0,0] => 1

[1,1,1,1,0,0,0,0,1,0] => 5

[1,1,1,1,0,0,0,1,0,0] => 1

[1,1,1,1,0,0,1,0,0,0] => 1

[1,1,1,1,0,1,0,0,0,0] => 1

[1,1,1,1,1,0,0,0,0,0] => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => 32

[1,0,1,0,1,0,1,0,1,1,0,0] => 24

[1,0,1,0,1,0,1,1,0,0,1,0] => 36

[1,0,1,0,1,0,1,1,0,1,0,0] => 12

[1,0,1,0,1,0,1,1,1,0,0,0] => 16

[1,0,1,0,1,1,0,0,1,0,1,0] => 36

[1,0,1,0,1,1,0,0,1,1,0,0] => 36

[1,0,1,0,1,1,0,1,0,0,1,0] => 18

[1,0,1,0,1,1,0,1,0,1,0,0] => 12

[1,0,1,0,1,1,0,1,1,0,0,0] => 6

[1,0,1,0,1,1,1,0,0,0,1,0] => 32

[1,0,1,0,1,1,1,0,0,1,0,0] => 8

[1,0,1,0,1,1,1,0,1,0,0,0] => 8

[1,0,1,0,1,1,1,1,0,0,0,0] => 10

[1,0,1,1,0,0,1,0,1,0,1,0] => 36

[1,0,1,1,0,0,1,0,1,1,0,0] => 27

[1,0,1,1,0,0,1,1,0,0,1,0] => 54

[1,0,1,1,0,0,1,1,0,1,0,0] => 18

[1,0,1,1,0,0,1,1,1,0,0,0] => 30

[1,0,1,1,0,1,0,0,1,0,1,0] => 18

[1,0,1,1,0,1,0,0,1,1,0,0] => 18

[1,0,1,1,0,1,0,1,0,0,1,0] => 18

[1,0,1,1,0,1,0,1,0,1,0,0] => 12

[1,0,1,1,0,1,0,1,1,0,0,0] => 9

[1,0,1,1,0,1,1,0,0,0,1,0] => 12

[1,0,1,1,0,1,1,0,0,1,0,0] => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => 6

[1,0,1,1,0,1,1,1,0,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => 32

[1,0,1,1,1,0,0,0,1,1,0,0] => 40

[1,0,1,1,1,0,0,1,0,0,1,0] => 12

[1,0,1,1,1,0,0,1,0,1,0,0] => 12

[1,0,1,1,1,0,0,1,1,0,0,0] => 4

[1,0,1,1,1,0,1,0,0,0,1,0] => 16

[1,0,1,1,1,0,1,0,0,1,0,0] => 8

[1,0,1,1,1,0,1,0,1,0,0,0] => 4

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Description

The number of 3-Catalan paths having this Dyck path as first and last coordinate projections.
A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$.
Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by projecting the Catalan path onto the $x,y$-plane and the $y,z$-plane, respectively.
For a given Dyck path $D$ this is the number of Catalan paths $C$ such that $D_{xy}(C) = D_{yz}(C) = D$.
If $D$ is of semilength $n$, $r_i(D)$ denotes the number of downsteps between the $i$-th and $(i+1)$-st upstep, and $s_i(D)$ denotes the number of upsteps between the $i$-th and $(i+1)$-st downstep, then this number is given by $\prod\limits_{i=1}^{n-1} \binom{r_i(D) + s_i(D)}{r_i(D)}$.

References

[1] Archer, K., Graves, C. Pattern-restricted permutations composed of 3-cycles MathSciNet:4400005 arXiv:2104.12664 DOI:10.1016/j.disc.2022.112895
[2] Archer, K., Gravies, C. A new statistic on Dyck paths for counting 3-dimensional Catalan words arXiv:2205.09686

Code

def statistic(D):
    # D is expected to be a list of 1s (upsteps) and 0s (downsteps)
    x = 1
    n = len(D) / 2
    uppies = [j for j, n in enumerate(D) if n == 1]
    downies = [j for j, n in enumerate(D) if n == 0]
    for i in range(1,n):
        r = D[ uppies[i-1] : uppies[i] ].count(0)
        s = D[ downies[i-1] : downies[i] ].count(1)
        x *= binomial(r + s, r)
    return x

Created

Jun 02, 2022 at 16:14 by Dennis Jahn

Updated

Jun 02, 2022 at 18:06 by Dennis Jahn