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.
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
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