Identifier
Values
=>
[1,1]=>3
[2]=>1
[1,1,1]=>6
[1,2]=>3
[2,1]=>1
[3]=>1
[1,1,1,1]=>10
[1,1,2]=>6
[1,2,1]=>3
[1,3]=>3
[2,1,1]=>1
[2,2]=>3
[3,1]=>1
[4]=>1
[1,1,1,1,1]=>15
[1,1,1,2]=>10
[1,1,2,1]=>6
[1,1,3]=>6
[1,2,1,1]=>3
[1,2,2]=>6
[1,3,1]=>3
[1,4]=>3
[2,1,1,1]=>1
[2,1,2]=>4
[2,2,1]=>3
[2,3]=>3
[3,1,1]=>1
[3,2]=>1
[4,1]=>1
[5]=>1
[1,1,1,1,1,1]=>21
[1,1,1,1,2]=>15
[1,1,1,2,1]=>10
[1,1,1,3]=>10
[1,1,2,1,1]=>6
[1,1,2,2]=>10
[1,1,3,1]=>6
[1,1,4]=>6
[1,2,1,1,1]=>3
[1,2,1,2]=>7
[1,2,2,1]=>6
[1,2,3]=>6
[1,3,1,1]=>3
[1,3,2]=>3
[1,4,1]=>3
[1,5]=>3
[2,1,1,1,1]=>1
[2,1,1,2]=>5
[2,1,2,1]=>4
[2,1,3]=>4
[2,2,1,1]=>3
[2,2,2]=>6
[2,3,1]=>3
[2,4]=>3
[3,1,1,1]=>1
[3,1,2]=>1
[3,2,1]=>1
[3,3]=>3
[4,1,1]=>1
[4,2]=>1
[5,1]=>1
[6]=>1
[1,1,1,1,1,1,1]=>28
[1,1,1,1,1,2]=>21
[1,1,1,1,2,1]=>15
[1,1,1,1,3]=>15
[1,1,1,2,1,1]=>10
[1,1,1,2,2]=>15
[1,1,1,3,1]=>10
[1,1,1,4]=>10
[1,1,2,1,1,1]=>6
[1,1,2,1,2]=>11
[1,1,2,2,1]=>10
[1,1,2,3]=>10
[1,1,3,1,1]=>6
[1,1,3,2]=>6
[1,1,4,1]=>6
[1,1,5]=>6
[1,2,1,1,1,1]=>3
[1,2,1,1,2]=>8
[1,2,1,2,1]=>7
[1,2,1,3]=>7
[1,2,2,1,1]=>6
[1,2,2,2]=>10
[1,2,3,1]=>6
[1,2,4]=>6
[1,3,1,1,1]=>3
[1,3,1,2]=>3
[1,3,2,1]=>3
[1,3,3]=>6
[1,4,1,1]=>3
[1,4,2]=>3
[1,5,1]=>3
[1,6]=>3
[2,1,1,1,1,1]=>1
[2,1,1,1,2]=>6
[2,1,1,2,1]=>5
[2,1,1,3]=>5
[2,1,2,1,1]=>4
[2,1,2,2]=>8
[2,1,3,1]=>4
[2,1,4]=>4
[2,2,1,1,1]=>3
[2,2,1,2]=>7
[2,2,2,1]=>6
[2,2,3]=>6
[2,3,1,1]=>3
[2,3,2]=>3
[2,4,1]=>3
[2,5]=>3
[3,1,1,1,1]=>1
[3,1,1,2]=>1
[3,1,2,1]=>1
[3,1,3]=>4
[3,2,1,1]=>1
[3,2,2]=>1
[3,3,1]=>3
[3,4]=>3
[4,1,1,1]=>1
[4,1,2]=>1
[4,2,1]=>1
[4,3]=>1
[5,1,1]=>1
[5,2]=>1
[6,1]=>1
[7]=>1
[1,1,1,1,1,1,1,1]=>36
[1,1,1,1,1,1,2]=>28
[1,1,1,1,1,2,1]=>21
[1,1,1,1,1,3]=>21
[1,1,1,1,2,1,1]=>15
[1,1,1,1,2,2]=>21
[1,1,1,1,3,1]=>15
[1,1,1,1,4]=>15
[1,1,1,2,1,1,1]=>10
[1,1,1,2,1,2]=>16
[1,1,1,2,2,1]=>15
[1,1,1,2,3]=>15
[1,1,1,3,1,1]=>10
[1,1,1,3,2]=>10
[1,1,1,4,1]=>10
[1,1,1,5]=>10
[1,1,2,1,1,1,1]=>6
[1,1,2,1,1,2]=>12
[1,1,2,1,2,1]=>11
[1,1,2,1,3]=>11
[1,1,2,2,1,1]=>10
[1,1,2,2,2]=>15
[1,1,2,3,1]=>10
[1,1,2,4]=>10
[1,1,3,1,1,1]=>6
[1,1,3,1,2]=>6
[1,1,3,2,1]=>6
[1,1,3,3]=>10
[1,1,4,1,1]=>6
[1,1,4,2]=>6
[1,1,5,1]=>6
[1,1,6]=>6
[1,2,1,1,1,1,1]=>3
[1,2,1,1,1,2]=>9
[1,2,1,1,2,1]=>8
[1,2,1,1,3]=>8
[1,2,1,2,1,1]=>7
[1,2,1,2,2]=>12
[1,2,1,3,1]=>7
[1,2,1,4]=>7
[1,2,2,1,1,1]=>6
[1,2,2,1,2]=>11
[1,2,2,2,1]=>10
[1,2,2,3]=>10
[1,2,3,1,1]=>6
[1,2,3,2]=>6
[1,2,4,1]=>6
[1,2,5]=>6
[1,3,1,1,1,1]=>3
[1,3,1,1,2]=>3
[1,3,1,2,1]=>3
[1,3,1,3]=>7
[1,3,2,1,1]=>3
[1,3,2,2]=>3
[1,3,3,1]=>6
[1,3,4]=>6
[1,4,1,1,1]=>3
[1,4,1,2]=>3
[1,4,2,1]=>3
[1,4,3]=>3
[1,5,1,1]=>3
[1,5,2]=>3
[1,6,1]=>3
[1,7]=>3
[2,1,1,1,1,1,1]=>1
[2,1,1,1,1,2]=>7
[2,1,1,1,2,1]=>6
[2,1,1,1,3]=>6
[2,1,1,2,1,1]=>5
[2,1,1,2,2]=>10
[2,1,1,3,1]=>5
[2,1,1,4]=>5
[2,1,2,1,1,1]=>4
[2,1,2,1,2]=>9
[2,1,2,2,1]=>8
[2,1,2,3]=>8
[2,1,3,1,1]=>4
[2,1,3,2]=>4
[2,1,4,1]=>4
[2,1,5]=>4
[2,2,1,1,1,1]=>3
[2,2,1,1,2]=>8
[2,2,1,2,1]=>7
[2,2,1,3]=>7
[2,2,2,1,1]=>6
[2,2,2,2]=>10
[2,2,3,1]=>6
[2,2,4]=>6
[2,3,1,1,1]=>3
[2,3,1,2]=>3
[2,3,2,1]=>3
[2,3,3]=>6
[2,4,1,1]=>3
[2,4,2]=>3
[2,5,1]=>3
[2,6]=>3
[3,1,1,1,1,1]=>1
[3,1,1,1,2]=>1
[3,1,1,2,1]=>1
[3,1,1,3]=>5
[3,1,2,1,1]=>1
[3,1,2,2]=>1
[3,1,3,1]=>4
[3,1,4]=>4
[3,2,1,1,1]=>1
[3,2,1,2]=>1
[3,2,2,1]=>1
[3,2,3]=>4
[3,3,1,1]=>3
[3,3,2]=>3
[3,4,1]=>3
[3,5]=>3
[4,1,1,1,1]=>1
[4,1,1,2]=>1
[4,1,2,1]=>1
[4,1,3]=>1
[4,2,1,1]=>1
[4,2,2]=>1
[4,3,1]=>1
[4,4]=>3
[5,1,1,1]=>1
[5,1,2]=>1
[5,2,1]=>1
[5,3]=>1
[6,1,1]=>1
[6,2]=>1
[7,1]=>1
[8]=>1
[1,1,1,1,1,1,1,1,1]=>45
[1,1,1,1,1,1,1,2]=>36
[1,1,1,1,1,1,2,1]=>28
[1,1,1,1,1,1,3]=>28
[1,1,1,1,1,2,1,1]=>21
[1,1,1,1,1,2,2]=>28
[1,1,1,1,1,3,1]=>21
[1,1,1,1,1,4]=>21
[1,1,1,1,2,1,1,1]=>15
[1,1,1,1,2,1,2]=>22
[1,1,1,1,2,2,1]=>21
[1,1,1,1,2,3]=>21
[1,1,1,1,3,1,1]=>15
[1,1,1,1,3,2]=>15
[1,1,1,1,4,1]=>15
[1,1,1,1,5]=>15
[1,1,1,2,1,1,1,1]=>10
[1,1,1,2,1,1,2]=>17
[1,1,1,2,1,2,1]=>16
[1,1,1,2,1,3]=>16
[1,1,1,2,2,1,1]=>15
[1,1,1,2,2,2]=>21
[1,1,1,2,3,1]=>15
[1,1,1,2,4]=>15
[1,1,1,3,1,1,1]=>10
[1,1,1,3,1,2]=>10
[1,1,1,3,2,1]=>10
[1,1,1,3,3]=>15
[1,1,1,4,1,1]=>10
[1,1,1,4,2]=>10
[1,1,1,5,1]=>10
[1,1,1,6]=>10
[1,1,2,1,1,1,1,1]=>6
[1,1,2,1,1,1,2]=>13
[1,1,2,1,1,2,1]=>12
[1,1,2,1,1,3]=>12
[1,1,2,1,2,1,1]=>11
[1,1,2,1,2,2]=>17
[1,1,2,1,3,1]=>11
[1,1,2,1,4]=>11
[1,1,2,2,1,1,1]=>10
[1,1,2,2,1,2]=>16
[1,1,2,2,2,1]=>15
[1,1,2,2,3]=>15
[1,1,2,3,1,1]=>10
[1,1,2,3,2]=>10
[1,1,2,4,1]=>10
[1,1,2,5]=>10
[1,1,3,1,1,1,1]=>6
[1,1,3,1,1,2]=>6
[1,1,3,1,2,1]=>6
[1,1,3,1,3]=>11
[1,1,3,2,1,1]=>6
[1,1,3,2,2]=>6
[1,1,3,3,1]=>10
[1,1,3,4]=>10
[1,1,4,1,1,1]=>6
[1,1,4,1,2]=>6
[1,1,4,2,1]=>6
[1,1,4,3]=>6
[1,1,5,1,1]=>6
[1,1,5,2]=>6
[1,1,6,1]=>6
[1,1,7]=>6
[1,2,1,1,1,1,1,1]=>3
[1,2,1,1,1,1,2]=>10
[1,2,1,1,1,2,1]=>9
[1,2,1,1,1,3]=>9
[1,2,1,1,2,1,1]=>8
[1,2,1,1,2,2]=>14
[1,2,1,1,3,1]=>8
[1,2,1,1,4]=>8
[1,2,1,2,1,1,1]=>7
[1,2,1,2,1,2]=>13
[1,2,1,2,2,1]=>12
[1,2,1,2,3]=>12
[1,2,1,3,1,1]=>7
[1,2,1,3,2]=>7
[1,2,1,4,1]=>7
[1,2,1,5]=>7
[1,2,2,1,1,1,1]=>6
[1,2,2,1,1,2]=>12
[1,2,2,1,2,1]=>11
[1,2,2,1,3]=>11
[1,2,2,2,1,1]=>10
[1,2,2,2,2]=>15
[1,2,2,3,1]=>10
[1,2,2,4]=>10
[1,2,3,1,1,1]=>6
[1,2,3,1,2]=>6
[1,2,3,2,1]=>6
[1,2,3,3]=>10
[1,2,4,1,1]=>6
[1,2,4,2]=>6
[1,2,5,1]=>6
[1,2,6]=>6
[1,3,1,1,1,1,1]=>3
[1,3,1,1,1,2]=>3
[1,3,1,1,2,1]=>3
[1,3,1,1,3]=>8
[1,3,1,2,1,1]=>3
[1,3,1,2,2]=>3
[1,3,1,3,1]=>7
[1,3,1,4]=>7
[1,3,2,1,1,1]=>3
[1,3,2,1,2]=>3
[1,3,2,2,1]=>3
[1,3,2,3]=>7
[1,3,3,1,1]=>6
[1,3,3,2]=>6
[1,3,4,1]=>6
[1,3,5]=>6
[1,4,1,1,1,1]=>3
[1,4,1,1,2]=>3
[1,4,1,2,1]=>3
[1,4,1,3]=>3
[1,4,2,1,1]=>3
[1,4,2,2]=>3
[1,4,3,1]=>3
[1,4,4]=>6
[1,5,1,1,1]=>3
[1,5,1,2]=>3
[1,5,2,1]=>3
[1,5,3]=>3
[1,6,1,1]=>3
[1,6,2]=>3
[1,7,1]=>3
[1,8]=>3
[2,1,1,1,1,1,1,1]=>1
[2,1,1,1,1,1,2]=>8
[2,1,1,1,1,2,1]=>7
[2,1,1,1,1,3]=>7
[2,1,1,1,2,1,1]=>6
[2,1,1,1,2,2]=>12
[2,1,1,1,3,1]=>6
[2,1,1,1,4]=>6
[2,1,1,2,1,1,1]=>5
[2,1,1,2,1,2]=>11
[2,1,1,2,2,1]=>10
[2,1,1,2,3]=>10
[2,1,1,3,1,1]=>5
[2,1,1,3,2]=>5
[2,1,1,4,1]=>5
[2,1,1,5]=>5
[2,1,2,1,1,1,1]=>4
[2,1,2,1,1,2]=>10
[2,1,2,1,2,1]=>9
[2,1,2,1,3]=>9
[2,1,2,2,1,1]=>8
[2,1,2,2,2]=>13
[2,1,2,3,1]=>8
[2,1,2,4]=>8
[2,1,3,1,1,1]=>4
[2,1,3,1,2]=>4
[2,1,3,2,1]=>4
[2,1,3,3]=>8
[2,1,4,1,1]=>4
[2,1,4,2]=>4
[2,1,5,1]=>4
[2,1,6]=>4
[2,2,1,1,1,1,1]=>3
[2,2,1,1,1,2]=>9
[2,2,1,1,2,1]=>8
[2,2,1,1,3]=>8
[2,2,1,2,1,1]=>7
[2,2,1,2,2]=>12
[2,2,1,3,1]=>7
[2,2,1,4]=>7
[2,2,2,1,1,1]=>6
[2,2,2,1,2]=>11
[2,2,2,2,1]=>10
[2,2,2,3]=>10
[2,2,3,1,1]=>6
[2,2,3,2]=>6
[2,2,4,1]=>6
[2,2,5]=>6
[2,3,1,1,1,1]=>3
[2,3,1,1,2]=>3
[2,3,1,2,1]=>3
[2,3,1,3]=>7
[2,3,2,1,1]=>3
[2,3,2,2]=>3
[2,3,3,1]=>6
[2,3,4]=>6
[2,4,1,1,1]=>3
[2,4,1,2]=>3
[2,4,2,1]=>3
[2,4,3]=>3
[2,5,1,1]=>3
[2,5,2]=>3
[2,6,1]=>3
[2,7]=>3
[3,1,1,1,1,1,1]=>1
[3,1,1,1,1,2]=>1
[3,1,1,1,2,1]=>1
[3,1,1,1,3]=>6
[3,1,1,2,1,1]=>1
[3,1,1,2,2]=>1
[3,1,1,3,1]=>5
[3,1,1,4]=>5
[3,1,2,1,1,1]=>1
[3,1,2,1,2]=>1
[3,1,2,2,1]=>1
[3,1,2,3]=>5
[3,1,3,1,1]=>4
[3,1,3,2]=>4
[3,1,4,1]=>4
[3,1,5]=>4
[3,2,1,1,1,1]=>1
[3,2,1,1,2]=>1
[3,2,1,2,1]=>1
[3,2,1,3]=>5
[3,2,2,1,1]=>1
[3,2,2,2]=>1
[3,2,3,1]=>4
[3,2,4]=>4
[3,3,1,1,1]=>3
[3,3,1,2]=>3
[3,3,2,1]=>3
[3,3,3]=>6
[3,4,1,1]=>3
[3,4,2]=>3
[3,5,1]=>3
[3,6]=>3
[4,1,1,1,1,1]=>1
[4,1,1,1,2]=>1
[4,1,1,2,1]=>1
[4,1,1,3]=>1
[4,1,2,1,1]=>1
[4,1,2,2]=>1
[4,1,3,1]=>1
[4,1,4]=>4
[4,2,1,1,1]=>1
[4,2,1,2]=>1
[4,2,2,1]=>1
[4,2,3]=>1
[4,3,1,1]=>1
[4,3,2]=>1
[4,4,1]=>3
[4,5]=>3
[5,1,1,1,1]=>1
[5,1,1,2]=>1
[5,1,2,1]=>1
[5,1,3]=>1
[5,2,1,1]=>1
[5,2,2]=>1
[5,3,1]=>1
[5,4]=>1
[6,1,1,1]=>1
[6,1,2]=>1
[6,2,1]=>1
[6,3]=>1
[7,1,1]=>1
[7,2]=>1
[8,1]=>1
[9]=>1
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Description
The sum of the positions of the weak records of an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$. This statistic is the sum of their positions.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$. This statistic is the sum of their positions.
References
[1] Knopfmacher, A., Mansour, T. Record statistics in integer compositions MathSciNet:2721540
Code
def statistic(pi):
pi = list(pi)
i = 0
res = 0
for j in range(len(pi)):
if pi[j] >= i:
i = pi[j]
res += j+1
return res
Created
Apr 08, 2017 at 22:01 by Martin Rubey
Updated
Apr 08, 2017 at 22:01 by Martin Rubey
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