- FindStat

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Identifier

St000394: Dyck paths ⟶ ℤ

Values

[1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + 4\ q + 9\ q^{2} + 12\ q^{3} + 10\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 25\ q^{3} + 31\ q^{4} + 26\ q^{5} + 16\ q^{6} + 9\ q^{7} + 4\ q^{8} + q^{9}$

$F_{7} = 1 + 6\ q + 20\ q^{2} + 44\ q^{3} + 70\ q^{4} + 82\ q^{5} + 74\ q^{6} + 54\ q^{7} + 38\ q^{8} + 22\ q^{9} + 12\ q^{10} + 4\ q^{11} + 2\ q^{12}$

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

References

[1] Feinberg, G., Lee, K.-H. Homogeneous representations of Type A KLR-algebras and Dyck paths arXiv:1505.04075
[2] Cheng, S.-E., Elizalde, S., Kasraoui, A., Sagan, B. Inversion polynomials for 321-avoiding permutations arXiv:1112.6014

Code

def statistic(D):
    heights = D.heights()
    return sum(heights[i+1]-1 for i in D.peaks())

Created

Feb 17, 2016 at 20:41 by Christian Stump

Updated

Dec 30, 2023 at 12:12 by Martin Rubey