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Identifier

St000645: Dyck paths ⟶ ℤ

Values

[] => 0

[1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,0,0,1,1,0,0] => 7

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

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

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

[1,0,1,0,1,1,1,0,0,0,1,0] => 7

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

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

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

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

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

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

[1,0,1,1,0,0,1,1,0,1,0,0] => 7

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

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

[1,0,1,1,0,1,0,0,1,1,0,0] => 7

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

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

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

[1,0,1,1,0,1,1,0,0,0,1,0] => 7

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

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

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

[1,0,1,1,1,0,0,0,1,0,1,0] => 7

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

[1,0,1,1,1,0,0,1,0,0,1,0] => 7

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

[1,0,1,1,1,0,0,1,1,0,0,0] => 7

[1,0,1,1,1,0,1,0,0,0,1,0] => 7

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

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

>>> Load all 2178 entries. <<<

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Description

The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$ \sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a) $$

Code

def statistic(D):
    I = D.peaks()
    J = D.valleys()
    return sum( (J[a]-I[a])*(I[a+1]-J[a]) for a in range(len(J)) )

Created

Nov 10, 2016 at 11:59 by Christian Stump

Updated

Apr 12, 2023 at 15:53 by Martin Rubey