- FindStat

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Identifier

St000675: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of centered multitunnels of a Dyck path.
This is the number of factorisations $D = A B C$ of a Dyck path, such that $B$ is a Dyck path and $A$ and $B$ have the same length.

References

[1] Elizalde, S., Deutsch, E. A simple and unusual bijection for Dyck paths and its consequences MathSciNet:2028229

Code

def statistic(D):
    """
    sage: D = DyckWord([1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0,0,1,0])
    sage: statistic(D)
    5
    """
    result = 0
    n = D.semilength()
    for i in range(1,n+1):
        T = D[n-i:n+i]
        if T.count(1) == i:
            try:
                DyckWord(T)
            except ValueError:
                pass
            else:
                result += 1
    return result

Created

Dec 27, 2016 at 00:10 by Martin Rubey

Updated

Dec 27, 2016 at 00:10 by Martin Rubey