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Identifier

St001034: Dyck paths ⟶ ℤ

Values

[] => 0

[1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 6

[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] => 6

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

[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] => 6

[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] => 6

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

[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] => 7

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

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

[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] => 6

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

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

[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] => 6

[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] => 6

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

[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] => 7

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

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

[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] => 7

[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] => 7

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

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

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

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

>>> Load all 2181 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = 4\ q^{3} + q^{4}$

$F_{4} = 8\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{5} = 16\ q^{5} + 12\ q^{6} + 9\ q^{7} + 4\ q^{8} + q^{9}$

$F_{6} = 32\ q^{6} + 32\ q^{7} + 30\ q^{8} + 20\ q^{9} + 12\ q^{10} + 4\ q^{11} + 2\ q^{12}$

$F_{7} = 64\ q^{7} + 80\ q^{8} + 88\ q^{9} + 73\ q^{10} + 56\ q^{11} + 34\ q^{12} + 20\ q^{13} + 9\ q^{14} + 4\ q^{15} + q^{16}$

Description

The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].

References

[1] Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the sum of the height of the peaks is k (n>=0; n<=k<=floor((n+1)^2/4)). OEIS:A129183

Code

# depends on https://trac.sagemath.org/ticket/16110
def statistic(D):
    return ParallelogramPolyomino.from_dyck_word(D).area()

Created

Nov 06, 2017 at 10:53 by Martin Rubey

Updated

Feb 05, 2022 at 21:55 by Martin Rubey