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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 [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