- FindStat

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Identifier

St001910: Dyck paths ⟶ ℤ

Values

[1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

[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] => 0

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

[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] => 3

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Description

The height of the middle non-run of a Dyck path.
A non-run of a Dyck path is a pair of steps which are not both up or both down, that is, which either form a peak or a valley. The number of non-runs is odd. This statistic returns the height of the middle non-run.

Code

def statistic(D):
    H = D.heights()
    T := [H[i+1] for i in range(len(D)-1) if D[i] != D[i+1]]
    return T[(len(T)-1)/2]

Created

Aug 07, 2023 at 18:04 by Martin Rubey

Updated

Aug 07, 2023 at 18:04 by Martin Rubey