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Identifier

St000808: Integer compositions ⟶ ℤ

Values

[1] => 1

[1,1] => 1

[2] => 2

[1,1,1] => 1

[1,2] => 2

[2,1] => 2

[3] => 3

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

[1,1,2] => 2

[1,2,1] => 2

[1,3] => 3

[2,1,1] => 2

[2,2] => 2

[3,1] => 3

[4] => 4

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

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

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

[1,1,3] => 3

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

[1,2,2] => 2

[1,3,1] => 3

[1,4] => 4

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

[2,1,2] => 3

[2,2,1] => 2

[2,3] => 3

[3,1,1] => 3

[3,2] => 3

[4,1] => 4

[5] => 5

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

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

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

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

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

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

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

[1,1,4] => 4

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

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

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

[1,2,3] => 3

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

[1,3,2] => 3

[1,4,1] => 4

[1,5] => 5

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

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

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

[2,1,3] => 4

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

[2,2,2] => 2

[2,3,1] => 3

[2,4] => 4

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

[3,1,2] => 4

[3,2,1] => 3

[3,3] => 3

[4,1,1] => 4

[4,2] => 4

[5,1] => 5

[6] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,5] => 5

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

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

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

[1,2,1,3] => 4

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

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

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

[1,2,4] => 4

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

[1,3,1,2] => 4

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

[1,3,3] => 3

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

[1,4,2] => 4

[1,5,1] => 5

[1,6] => 6

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

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

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

[2,1,1,3] => 4

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

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

>>> Load all 511 entries. <<<

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Description

The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.

References

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

Code

def statistic(c):
    if len(c) == 0:
        return 0
    U = c[0]
    for i in range(1, len(c)):
        if c[i] > c[i-1]:
            U += c[i]-c[i-1]
    return U

Created

May 18, 2017 at 09:45 by Martin Rubey

Updated

Nov 19, 2017 at 22:04 by Christian Stump