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Identifier

St000805: Integer compositions ⟶ ℤ

Values

[1] => 1

[1,1] => 1

[2] => 1

[1,1,1] => 1

[1,2] => 1

[2,1] => 1

[3] => 1

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

[1,1,2] => 1

[1,2,1] => 1

[1,3] => 1

[2,1,1] => 1

[2,2] => 1

[3,1] => 1

[4] => 1

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

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

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

[1,1,3] => 1

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

[1,2,2] => 1

[1,3,1] => 1

[1,4] => 1

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

[2,1,2] => 2

[2,2,1] => 1

[2,3] => 1

[3,1,1] => 1

[3,2] => 1

[4,1] => 1

[5] => 1

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

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

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

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

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

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

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

[1,1,4] => 1

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

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

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

[1,2,3] => 1

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

[1,3,2] => 1

[1,4,1] => 1

[1,5] => 1

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

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

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

[2,1,3] => 2

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

[2,2,2] => 1

[2,3,1] => 1

[2,4] => 1

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

[3,1,2] => 2

[3,2,1] => 1

[3,3] => 1

[4,1,1] => 1

[4,2] => 1

[5,1] => 1

[6] => 1

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

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

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

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

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

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

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

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

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

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

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

[1,1,2,3] => 1

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

[1,1,3,2] => 1

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

[1,1,5] => 1

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

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

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

[1,2,1,3] => 2

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

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

[1,2,3,1] => 1

[1,2,4] => 1

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

[1,3,1,2] => 2

[1,3,2,1] => 1

[1,3,3] => 1

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

[1,4,2] => 1

[1,5,1] => 1

[1,6] => 1

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

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

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

[2,1,1,3] => 2

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

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

>>> Load all 511 entries. <<<

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Description

The number of peaks of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of contiguous subsequences consisting of an up step, a sequence of horizontal steps, and a down step.

References

[1] Deutsch, E., Elizalde, S. A bijection between bargraphs and Dyck paths arXiv:1705.05984

Code

def statistic(c):
    c = [0] + list(c) + [0]
    pk = 0
    found_UH = False
    for i in range(1,len(c)):
        if found_UH:
            if c[i] < c[i-1]:
                found_UH = False
                pk += 1
        else:
            if c[i] > c[i-1]:
                found_UH = True
    return pk

Created

May 18, 2017 at 09:17 by Martin Rubey

Updated

Nov 19, 2017 at 22:05 by Christian Stump