- FindStat

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Identifier

St000674: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

[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] => 3

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

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

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

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

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

[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] => 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] => 1

[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] => 2

[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] => 1

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

[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] => 2

[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] => 1

[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] => 1

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

[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] => 2

>>> Load all 1200 entries. <<<

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Description

The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.

References

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

Code

def statistic(D):
    H = D.heights()
    return len([p for p in D.peaks() if H[p] == 0])

Created

Dec 26, 2016 at 23:37 by Martin Rubey

Updated

Dec 26, 2016 at 23:37 by Martin Rubey