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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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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + 3\ q^{2}$

$F_{4} = 1 + q + 3\ q^{2} + 7\ q^{3} + 2\ q^{4}$

$F_{5} = 1 + q + 3\ q^{2} + 7\ q^{3} + 18\ q^{4} + 7\ q^{5} + 5\ q^{6}$

$F_{6} = 1 + q + 3\ q^{2} + 7\ q^{3} + 18\ q^{4} + 43\ q^{5} + 28\ q^{6} + 19\ q^{7} + 9\ q^{8} + 3\ q^{9}$

$F_{7} = 1 + q + 3\ q^{2} + 7\ q^{3} + 18\ q^{4} + 43\ q^{5} + 108\ q^{6} + 87\ q^{7} + 79\ q^{8} + 41\ q^{9} + 28\ q^{10} + 6\ q^{11} + 7\ q^{12}$

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