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Identifier
Values
[1,1] => 3
[2] => 3
[1,1,1] => 4
[1,2] => 4
[2,1] => 4
[3] => 4
[1,1,1,1] => 5
[1,1,2] => 5
[1,2,1] => 5
[1,3] => 5
[2,1,1] => 5
[2,2] => 4
[3,1] => 5
[4] => 5
[1,1,1,1,1] => 6
[1,1,1,2] => 6
[1,1,2,1] => 6
[1,1,3] => 6
[1,2,1,1] => 6
[1,2,2] => 5
[1,3,1] => 6
[1,4] => 6
[2,1,1,1] => 6
[2,1,2] => 6
[2,2,1] => 5
[2,3] => 5
[3,1,1] => 6
[3,2] => 5
[4,1] => 6
[5] => 6
[1,1,1,1,1,1] => 7
[1,1,1,1,2] => 7
[1,1,1,2,1] => 7
[1,1,1,3] => 7
[1,1,2,1,1] => 7
[1,1,2,2] => 6
[1,1,3,1] => 7
[1,1,4] => 7
[1,2,1,1,1] => 7
[1,2,1,2] => 7
[1,2,2,1] => 6
[1,2,3] => 6
[1,3,1,1] => 7
[1,3,2] => 6
[1,4,1] => 7
[1,5] => 7
[2,1,1,1,1] => 7
[2,1,1,2] => 7
[2,1,2,1] => 7
[2,1,3] => 7
[2,2,1,1] => 6
[2,2,2] => 5
[2,3,1] => 6
[2,4] => 6
[3,1,1,1] => 7
[3,1,2] => 7
[3,2,1] => 6
[3,3] => 5
[4,1,1] => 7
[4,2] => 6
[5,1] => 7
[6] => 7
[1,1,1,1,1,1,1] => 8
[1,1,1,1,1,2] => 8
[1,1,1,1,2,1] => 8
[1,1,1,1,3] => 8
[1,1,1,2,1,1] => 8
[1,1,1,2,2] => 7
[1,1,1,3,1] => 8
[1,1,1,4] => 8
[1,1,2,1,1,1] => 8
[1,1,2,1,2] => 8
[1,1,2,2,1] => 7
[1,1,2,3] => 7
[1,1,3,1,1] => 8
[1,1,3,2] => 7
[1,1,4,1] => 8
[1,1,5] => 8
[1,2,1,1,1,1] => 8
[1,2,1,1,2] => 8
[1,2,1,2,1] => 8
[1,2,1,3] => 8
[1,2,2,1,1] => 7
[1,2,2,2] => 6
[1,2,3,1] => 7
[1,2,4] => 7
[1,3,1,1,1] => 8
[1,3,1,2] => 8
[1,3,2,1] => 7
[1,3,3] => 6
[1,4,1,1] => 8
[1,4,2] => 7
[1,5,1] => 8
[1,6] => 8
[2,1,1,1,1,1] => 8
[2,1,1,1,2] => 8
[2,1,1,2,1] => 8
[2,1,1,3] => 8
[2,1,2,1,1] => 8
[2,1,2,2] => 7
[2,1,3,1] => 8
>>> Load all 510 entries. <<<
[2,1,4] => 8
[2,2,1,1,1] => 7
[2,2,1,2] => 7
[2,2,2,1] => 6
[2,2,3] => 6
[2,3,1,1] => 7
[2,3,2] => 6
[2,4,1] => 7
[2,5] => 7
[3,1,1,1,1] => 8
[3,1,1,2] => 8
[3,1,2,1] => 8
[3,1,3] => 8
[3,2,1,1] => 7
[3,2,2] => 6
[3,3,1] => 6
[3,4] => 6
[4,1,1,1] => 8
[4,1,2] => 8
[4,2,1] => 7
[4,3] => 6
[5,1,1] => 8
[5,2] => 7
[6,1] => 8
[7] => 8
[1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,2] => 9
[1,1,1,1,1,2,1] => 9
[1,1,1,1,1,3] => 9
[1,1,1,1,2,1,1] => 9
[1,1,1,1,2,2] => 8
[1,1,1,1,3,1] => 9
[1,1,1,1,4] => 9
[1,1,1,2,1,1,1] => 9
[1,1,1,2,1,2] => 9
[1,1,1,2,2,1] => 8
[1,1,1,2,3] => 8
[1,1,1,3,1,1] => 9
[1,1,1,3,2] => 8
[1,1,1,4,1] => 9
[1,1,1,5] => 9
[1,1,2,1,1,1,1] => 9
[1,1,2,1,1,2] => 9
[1,1,2,1,2,1] => 9
[1,1,2,1,3] => 9
[1,1,2,2,1,1] => 8
[1,1,2,2,2] => 7
[1,1,2,3,1] => 8
[1,1,2,4] => 8
[1,1,3,1,1,1] => 9
[1,1,3,1,2] => 9
[1,1,3,2,1] => 8
[1,1,3,3] => 7
[1,1,4,1,1] => 9
[1,1,4,2] => 8
[1,1,5,1] => 9
[1,1,6] => 9
[1,2,1,1,1,1,1] => 9
[1,2,1,1,1,2] => 9
[1,2,1,1,2,1] => 9
[1,2,1,1,3] => 9
[1,2,1,2,1,1] => 9
[1,2,1,2,2] => 8
[1,2,1,3,1] => 9
[1,2,1,4] => 9
[1,2,2,1,1,1] => 8
[1,2,2,1,2] => 8
[1,2,2,2,1] => 7
[1,2,2,3] => 7
[1,2,3,1,1] => 8
[1,2,3,2] => 7
[1,2,4,1] => 8
[1,2,5] => 8
[1,3,1,1,1,1] => 9
[1,3,1,1,2] => 9
[1,3,1,2,1] => 9
[1,3,1,3] => 9
[1,3,2,1,1] => 8
[1,3,2,2] => 7
[1,3,3,1] => 7
[1,3,4] => 7
[1,4,1,1,1] => 9
[1,4,1,2] => 9
[1,4,2,1] => 8
[1,4,3] => 7
[1,5,1,1] => 9
[1,5,2] => 8
[1,6,1] => 9
[1,7] => 9
[2,1,1,1,1,1,1] => 9
[2,1,1,1,1,2] => 9
[2,1,1,1,2,1] => 9
[2,1,1,1,3] => 9
[2,1,1,2,1,1] => 9
[2,1,1,2,2] => 8
[2,1,1,3,1] => 9
[2,1,1,4] => 9
[2,1,2,1,1,1] => 9
[2,1,2,1,2] => 9
[2,1,2,2,1] => 8
[2,1,2,3] => 8
[2,1,3,1,1] => 9
[2,1,3,2] => 8
[2,1,4,1] => 9
[2,1,5] => 9
[2,2,1,1,1,1] => 8
[2,2,1,1,2] => 8
[2,2,1,2,1] => 8
[2,2,1,3] => 8
[2,2,2,1,1] => 7
[2,2,2,2] => 6
[2,2,3,1] => 7
[2,2,4] => 7
[2,3,1,1,1] => 8
[2,3,1,2] => 8
[2,3,2,1] => 7
[2,3,3] => 6
[2,4,1,1] => 8
[2,4,2] => 7
[2,5,1] => 8
[2,6] => 8
[3,1,1,1,1,1] => 9
[3,1,1,1,2] => 9
[3,1,1,2,1] => 9
[3,1,1,3] => 9
[3,1,2,1,1] => 9
[3,1,2,2] => 8
[3,1,3,1] => 9
[3,1,4] => 9
[3,2,1,1,1] => 8
[3,2,1,2] => 8
[3,2,2,1] => 7
[3,2,3] => 7
[3,3,1,1] => 7
[3,3,2] => 6
[3,4,1] => 7
[3,5] => 7
[4,1,1,1,1] => 9
[4,1,1,2] => 9
[4,1,2,1] => 9
[4,1,3] => 9
[4,2,1,1] => 8
[4,2,2] => 7
[4,3,1] => 7
[4,4] => 6
[5,1,1,1] => 9
[5,1,2] => 9
[5,2,1] => 8
[5,3] => 7
[6,1,1] => 9
[6,2] => 8
[7,1] => 9
[8] => 9
[1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,2] => 10
[1,1,1,1,1,1,2,1] => 10
[1,1,1,1,1,1,3] => 10
[1,1,1,1,1,2,1,1] => 10
[1,1,1,1,1,2,2] => 9
[1,1,1,1,1,3,1] => 10
[1,1,1,1,1,4] => 10
[1,1,1,1,2,1,1,1] => 10
[1,1,1,1,2,1,2] => 10
[1,1,1,1,2,2,1] => 9
[1,1,1,1,2,3] => 9
[1,1,1,1,3,1,1] => 10
[1,1,1,1,3,2] => 9
[1,1,1,1,4,1] => 10
[1,1,1,1,5] => 10
[1,1,1,2,1,1,1,1] => 10
[1,1,1,2,1,1,2] => 10
[1,1,1,2,1,2,1] => 10
[1,1,1,2,1,3] => 10
[1,1,1,2,2,1,1] => 9
[1,1,1,2,2,2] => 8
[1,1,1,2,3,1] => 9
[1,1,1,2,4] => 9
[1,1,1,3,1,1,1] => 10
[1,1,1,3,1,2] => 10
[1,1,1,3,2,1] => 9
[1,1,1,3,3] => 8
[1,1,1,4,1,1] => 10
[1,1,1,4,2] => 9
[1,1,1,5,1] => 10
[1,1,1,6] => 10
[1,1,2,1,1,1,1,1] => 10
[1,1,2,1,1,1,2] => 10
[1,1,2,1,1,2,1] => 10
[1,1,2,1,1,3] => 10
[1,1,2,1,2,1,1] => 10
[1,1,2,1,2,2] => 9
[1,1,2,1,3,1] => 10
[1,1,2,1,4] => 10
[1,1,2,2,1,1,1] => 9
[1,1,2,2,1,2] => 9
[1,1,2,2,2,1] => 8
[1,1,2,2,3] => 8
[1,1,2,3,1,1] => 9
[1,1,2,3,2] => 8
[1,1,2,4,1] => 9
[1,1,2,5] => 9
[1,1,3,1,1,1,1] => 10
[1,1,3,1,1,2] => 10
[1,1,3,1,2,1] => 10
[1,1,3,1,3] => 10
[1,1,3,2,1,1] => 9
[1,1,3,2,2] => 8
[1,1,3,3,1] => 8
[1,1,3,4] => 8
[1,1,4,1,1,1] => 10
[1,1,4,1,2] => 10
[1,1,4,2,1] => 9
[1,1,4,3] => 8
[1,1,5,1,1] => 10
[1,1,5,2] => 9
[1,1,6,1] => 10
[1,1,7] => 10
[1,2,1,1,1,1,1,1] => 10
[1,2,1,1,1,1,2] => 10
[1,2,1,1,1,2,1] => 10
[1,2,1,1,1,3] => 10
[1,2,1,1,2,1,1] => 10
[1,2,1,1,2,2] => 9
[1,2,1,1,3,1] => 10
[1,2,1,1,4] => 10
[1,2,1,2,1,1,1] => 10
[1,2,1,2,1,2] => 10
[1,2,1,2,2,1] => 9
[1,2,1,2,3] => 9
[1,2,1,3,1,1] => 10
[1,2,1,3,2] => 9
[1,2,1,4,1] => 10
[1,2,1,5] => 10
[1,2,2,1,1,1,1] => 9
[1,2,2,1,1,2] => 9
[1,2,2,1,2,1] => 9
[1,2,2,1,3] => 9
[1,2,2,2,1,1] => 8
[1,2,2,2,2] => 7
[1,2,2,3,1] => 8
[1,2,2,4] => 8
[1,2,3,1,1,1] => 9
[1,2,3,1,2] => 9
[1,2,3,2,1] => 8
[1,2,3,3] => 7
[1,2,4,1,1] => 9
[1,2,4,2] => 8
[1,2,5,1] => 9
[1,2,6] => 9
[1,3,1,1,1,1,1] => 10
[1,3,1,1,1,2] => 10
[1,3,1,1,2,1] => 10
[1,3,1,1,3] => 10
[1,3,1,2,1,1] => 10
[1,3,1,2,2] => 9
[1,3,1,3,1] => 10
[1,3,1,4] => 10
[1,3,2,1,1,1] => 9
[1,3,2,1,2] => 9
[1,3,2,2,1] => 8
[1,3,2,3] => 8
[1,3,3,1,1] => 8
[1,3,3,2] => 7
[1,3,4,1] => 8
[1,3,5] => 8
[1,4,1,1,1,1] => 10
[1,4,1,1,2] => 10
[1,4,1,2,1] => 10
[1,4,1,3] => 10
[1,4,2,1,1] => 9
[1,4,2,2] => 8
[1,4,3,1] => 8
[1,4,4] => 7
[1,5,1,1,1] => 10
[1,5,1,2] => 10
[1,5,2,1] => 9
[1,5,3] => 8
[1,6,1,1] => 10
[1,6,2] => 9
[1,7,1] => 10
[1,8] => 10
[2,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,2] => 10
[2,1,1,1,1,2,1] => 10
[2,1,1,1,1,3] => 10
[2,1,1,1,2,1,1] => 10
[2,1,1,1,2,2] => 9
[2,1,1,1,3,1] => 10
[2,1,1,1,4] => 10
[2,1,1,2,1,1,1] => 10
[2,1,1,2,1,2] => 10
[2,1,1,2,2,1] => 9
[2,1,1,2,3] => 9
[2,1,1,3,1,1] => 10
[2,1,1,3,2] => 9
[2,1,1,4,1] => 10
[2,1,1,5] => 10
[2,1,2,1,1,1,1] => 10
[2,1,2,1,1,2] => 10
[2,1,2,1,2,1] => 10
[2,1,2,1,3] => 10
[2,1,2,2,1,1] => 9
[2,1,2,2,2] => 8
[2,1,2,3,1] => 9
[2,1,2,4] => 9
[2,1,3,1,1,1] => 10
[2,1,3,1,2] => 10
[2,1,3,2,1] => 9
[2,1,3,3] => 8
[2,1,4,1,1] => 10
[2,1,4,2] => 9
[2,1,5,1] => 10
[2,1,6] => 10
[2,2,1,1,1,1,1] => 9
[2,2,1,1,1,2] => 9
[2,2,1,1,2,1] => 9
[2,2,1,1,3] => 9
[2,2,1,2,1,1] => 9
[2,2,1,2,2] => 8
[2,2,1,3,1] => 9
[2,2,1,4] => 9
[2,2,2,1,1,1] => 8
[2,2,2,1,2] => 8
[2,2,2,2,1] => 7
[2,2,2,3] => 7
[2,2,3,1,1] => 8
[2,2,3,2] => 7
[2,2,4,1] => 8
[2,2,5] => 8
[2,3,1,1,1,1] => 9
[2,3,1,1,2] => 9
[2,3,1,2,1] => 9
[2,3,1,3] => 9
[2,3,2,1,1] => 8
[2,3,2,2] => 7
[2,3,3,1] => 7
[2,3,4] => 7
[2,4,1,1,1] => 9
[2,4,1,2] => 9
[2,4,2,1] => 8
[2,4,3] => 7
[2,5,1,1] => 9
[2,5,2] => 8
[2,6,1] => 9
[2,7] => 9
[3,1,1,1,1,1,1] => 10
[3,1,1,1,1,2] => 10
[3,1,1,1,2,1] => 10
[3,1,1,1,3] => 10
[3,1,1,2,1,1] => 10
[3,1,1,2,2] => 9
[3,1,1,3,1] => 10
[3,1,1,4] => 10
[3,1,2,1,1,1] => 10
[3,1,2,1,2] => 10
[3,1,2,2,1] => 9
[3,1,2,3] => 9
[3,1,3,1,1] => 10
[3,1,3,2] => 9
[3,1,4,1] => 10
[3,1,5] => 10
[3,2,1,1,1,1] => 9
[3,2,1,1,2] => 9
[3,2,1,2,1] => 9
[3,2,1,3] => 9
[3,2,2,1,1] => 8
[3,2,2,2] => 7
[3,2,3,1] => 8
[3,2,4] => 8
[3,3,1,1,1] => 8
[3,3,1,2] => 8
[3,3,2,1] => 7
[3,3,3] => 6
[3,4,1,1] => 8
[3,4,2] => 7
[3,5,1] => 8
[3,6] => 8
[4,1,1,1,1,1] => 10
[4,1,1,1,2] => 10
[4,1,1,2,1] => 10
[4,1,1,3] => 10
[4,1,2,1,1] => 10
[4,1,2,2] => 9
[4,1,3,1] => 10
[4,1,4] => 10
[4,2,1,1,1] => 9
[4,2,1,2] => 9
[4,2,2,1] => 8
[4,2,3] => 8
[4,3,1,1] => 8
[4,3,2] => 7
[4,4,1] => 7
[4,5] => 7
[5,1,1,1,1] => 10
[5,1,1,2] => 10
[5,1,2,1] => 10
[5,1,3] => 10
[5,2,1,1] => 9
[5,2,2] => 8
[5,3,1] => 8
[5,4] => 7
[6,1,1,1] => 10
[6,1,2] => 10
[6,2,1] => 9
[6,3] => 8
[7,1,1] => 10
[7,2] => 9
[8,1] => 10
[9] => 10
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Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down 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
    sp = 1+c[0]
    for i in range(1, len(c)):
        if c[i] > c[i-1]:
            sp += 1+c[i]-c[i-1]
        else:
            sp += 1
    return sp
Created
May 18, 2017 at 08:32 by Martin Rubey
Updated
May 22, 2017 at 07:22 by Martin Rubey