Identifier
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
click to show generating function       
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.
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