- FindStat

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Identifier

St000931: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

[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] => 0

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

[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] => 0

[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] => 1

[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] => 1

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

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

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

[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] => 2

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

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

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

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

[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] => 0

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

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

[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] => 0

[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] => 0

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

[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] => 1

[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] => 2

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

[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] => 1

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

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

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

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

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

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

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

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

[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] => 1

[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] => 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] => 1

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

[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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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 4\ q + q^{2}$

$F_{5} = 21 + 15\ q + 5\ q^{2} + q^{3}$

$F_{6} = 51 + 50\ q + 24\ q^{2} + 6\ q^{3} + q^{4}$

$F_{7} = 127 + 161\ q + 98\ q^{2} + 35\ q^{3} + 7\ q^{4} + q^{5}$

Description

The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].

References

[1] Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle. OEIS:A001006

Code

def statistic(D):
    return sum(1 for i in range(len(D)-2) if D[i:i+3] == [1,1,1])

Created

Aug 09, 2017 at 14:30 by Christian Stump

Updated

Aug 09, 2017 at 14:30 by Christian Stump