edit this statistic or download as text // json
Identifier
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 1
[2,1] => 0
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 0
[2,2] => 0
[3,1] => 0
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 0
[2,1,2] => 1
[2,2,1] => 0
[2,3] => 1
[3,1,1] => 0
[3,2] => 0
[4,1] => 0
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 2
[1,2,2,1] => 1
[1,2,3] => 2
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 0
[2,1,1,2] => 1
[2,1,2,1] => 1
[2,1,3] => 1
[2,2,1,1] => 0
[2,2,2] => 0
[2,3,1] => 1
[2,4] => 1
[3,1,1,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[3,3] => 0
[4,1,1] => 0
[4,2] => 0
[5,1] => 0
[6] => 0
[1,1,1,1,1,1,1] => 0
[1,1,1,1,1,2] => 1
[1,1,1,1,2,1] => 1
[1,1,1,1,3] => 1
[1,1,1,2,1,1] => 1
[1,1,1,2,2] => 1
[1,1,1,3,1] => 1
[1,1,1,4] => 1
[1,1,2,1,1,1] => 1
[1,1,2,1,2] => 2
[1,1,2,2,1] => 1
[1,1,2,3] => 2
[1,1,3,1,1] => 1
[1,1,3,2] => 1
[1,1,4,1] => 1
[1,1,5] => 1
[1,2,1,1,1,1] => 1
[1,2,1,1,2] => 2
[1,2,1,2,1] => 2
[1,2,1,3] => 2
[1,2,2,1,1] => 1
[1,2,2,2] => 1
[1,2,3,1] => 2
[1,2,4] => 2
[1,3,1,1,1] => 1
[1,3,1,2] => 2
[1,3,2,1] => 1
[1,3,3] => 1
[1,4,1,1] => 1
[1,4,2] => 1
[1,5,1] => 1
[1,6] => 1
[2,1,1,1,1,1] => 0
[2,1,1,1,2] => 1
[2,1,1,2,1] => 1
[2,1,1,3] => 1
[2,1,2,1,1] => 1
[2,1,2,2] => 1
>>> Load all 511 entries. <<<
[2,1,3,1] => 1
[2,1,4] => 1
[2,2,1,1,1] => 0
[2,2,1,2] => 1
[2,2,2,1] => 0
[2,2,3] => 1
[2,3,1,1] => 1
[2,3,2] => 1
[2,4,1] => 1
[2,5] => 1
[3,1,1,1,1] => 0
[3,1,1,2] => 1
[3,1,2,1] => 1
[3,1,3] => 1
[3,2,1,1] => 0
[3,2,2] => 0
[3,3,1] => 0
[3,4] => 1
[4,1,1,1] => 0
[4,1,2] => 1
[4,2,1] => 0
[4,3] => 0
[5,1,1] => 0
[5,2] => 0
[6,1] => 0
[7] => 0
[1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,2] => 1
[1,1,1,1,1,2,1] => 1
[1,1,1,1,1,3] => 1
[1,1,1,1,2,1,1] => 1
[1,1,1,1,2,2] => 1
[1,1,1,1,3,1] => 1
[1,1,1,1,4] => 1
[1,1,1,2,1,1,1] => 1
[1,1,1,2,1,2] => 2
[1,1,1,2,2,1] => 1
[1,1,1,2,3] => 2
[1,1,1,3,1,1] => 1
[1,1,1,3,2] => 1
[1,1,1,4,1] => 1
[1,1,1,5] => 1
[1,1,2,1,1,1,1] => 1
[1,1,2,1,1,2] => 2
[1,1,2,1,2,1] => 2
[1,1,2,1,3] => 2
[1,1,2,2,1,1] => 1
[1,1,2,2,2] => 1
[1,1,2,3,1] => 2
[1,1,2,4] => 2
[1,1,3,1,1,1] => 1
[1,1,3,1,2] => 2
[1,1,3,2,1] => 1
[1,1,3,3] => 1
[1,1,4,1,1] => 1
[1,1,4,2] => 1
[1,1,5,1] => 1
[1,1,6] => 1
[1,2,1,1,1,1,1] => 1
[1,2,1,1,1,2] => 2
[1,2,1,1,2,1] => 2
[1,2,1,1,3] => 2
[1,2,1,2,1,1] => 2
[1,2,1,2,2] => 2
[1,2,1,3,1] => 2
[1,2,1,4] => 2
[1,2,2,1,1,1] => 1
[1,2,2,1,2] => 2
[1,2,2,2,1] => 1
[1,2,2,3] => 2
[1,2,3,1,1] => 2
[1,2,3,2] => 2
[1,2,4,1] => 2
[1,2,5] => 2
[1,3,1,1,1,1] => 1
[1,3,1,1,2] => 2
[1,3,1,2,1] => 2
[1,3,1,3] => 2
[1,3,2,1,1] => 1
[1,3,2,2] => 1
[1,3,3,1] => 1
[1,3,4] => 2
[1,4,1,1,1] => 1
[1,4,1,2] => 2
[1,4,2,1] => 1
[1,4,3] => 1
[1,5,1,1] => 1
[1,5,2] => 1
[1,6,1] => 1
[1,7] => 1
[2,1,1,1,1,1,1] => 0
[2,1,1,1,1,2] => 1
[2,1,1,1,2,1] => 1
[2,1,1,1,3] => 1
[2,1,1,2,1,1] => 1
[2,1,1,2,2] => 1
[2,1,1,3,1] => 1
[2,1,1,4] => 1
[2,1,2,1,1,1] => 1
[2,1,2,1,2] => 2
[2,1,2,2,1] => 1
[2,1,2,3] => 2
[2,1,3,1,1] => 1
[2,1,3,2] => 1
[2,1,4,1] => 1
[2,1,5] => 1
[2,2,1,1,1,1] => 0
[2,2,1,1,2] => 1
[2,2,1,2,1] => 1
[2,2,1,3] => 1
[2,2,2,1,1] => 0
[2,2,2,2] => 0
[2,2,3,1] => 1
[2,2,4] => 1
[2,3,1,1,1] => 1
[2,3,1,2] => 2
[2,3,2,1] => 1
[2,3,3] => 1
[2,4,1,1] => 1
[2,4,2] => 1
[2,5,1] => 1
[2,6] => 1
[3,1,1,1,1,1] => 0
[3,1,1,1,2] => 1
[3,1,1,2,1] => 1
[3,1,1,3] => 1
[3,1,2,1,1] => 1
[3,1,2,2] => 1
[3,1,3,1] => 1
[3,1,4] => 1
[3,2,1,1,1] => 0
[3,2,1,2] => 1
[3,2,2,1] => 0
[3,2,3] => 1
[3,3,1,1] => 0
[3,3,2] => 0
[3,4,1] => 1
[3,5] => 1
[4,1,1,1,1] => 0
[4,1,1,2] => 1
[4,1,2,1] => 1
[4,1,3] => 1
[4,2,1,1] => 0
[4,2,2] => 0
[4,3,1] => 0
[4,4] => 0
[5,1,1,1] => 0
[5,1,2] => 1
[5,2,1] => 0
[5,3] => 0
[6,1,1] => 0
[6,2] => 0
[7,1] => 0
[8] => 0
[1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,2] => 1
[1,1,1,1,1,1,2,1] => 1
[1,1,1,1,1,1,3] => 1
[1,1,1,1,1,2,1,1] => 1
[1,1,1,1,1,2,2] => 1
[1,1,1,1,1,3,1] => 1
[1,1,1,1,1,4] => 1
[1,1,1,1,2,1,1,1] => 1
[1,1,1,1,2,1,2] => 2
[1,1,1,1,2,2,1] => 1
[1,1,1,1,2,3] => 2
[1,1,1,1,3,1,1] => 1
[1,1,1,1,3,2] => 1
[1,1,1,1,4,1] => 1
[1,1,1,1,5] => 1
[1,1,1,2,1,1,1,1] => 1
[1,1,1,2,1,1,2] => 2
[1,1,1,2,1,2,1] => 2
[1,1,1,2,1,3] => 2
[1,1,1,2,2,1,1] => 1
[1,1,1,2,2,2] => 1
[1,1,1,2,3,1] => 2
[1,1,1,2,4] => 2
[1,1,1,3,1,1,1] => 1
[1,1,1,3,1,2] => 2
[1,1,1,3,2,1] => 1
[1,1,1,3,3] => 1
[1,1,1,4,1,1] => 1
[1,1,1,4,2] => 1
[1,1,1,5,1] => 1
[1,1,1,6] => 1
[1,1,2,1,1,1,1,1] => 1
[1,1,2,1,1,1,2] => 2
[1,1,2,1,1,2,1] => 2
[1,1,2,1,1,3] => 2
[1,1,2,1,2,1,1] => 2
[1,1,2,1,2,2] => 2
[1,1,2,1,3,1] => 2
[1,1,2,1,4] => 2
[1,1,2,2,1,1,1] => 1
[1,1,2,2,1,2] => 2
[1,1,2,2,2,1] => 1
[1,1,2,2,3] => 2
[1,1,2,3,1,1] => 2
[1,1,2,3,2] => 2
[1,1,2,4,1] => 2
[1,1,2,5] => 2
[1,1,3,1,1,1,1] => 1
[1,1,3,1,1,2] => 2
[1,1,3,1,2,1] => 2
[1,1,3,1,3] => 2
[1,1,3,2,1,1] => 1
[1,1,3,2,2] => 1
[1,1,3,3,1] => 1
[1,1,3,4] => 2
[1,1,4,1,1,1] => 1
[1,1,4,1,2] => 2
[1,1,4,2,1] => 1
[1,1,4,3] => 1
[1,1,5,1,1] => 1
[1,1,5,2] => 1
[1,1,6,1] => 1
[1,1,7] => 1
[1,2,1,1,1,1,1,1] => 1
[1,2,1,1,1,1,2] => 2
[1,2,1,1,1,2,1] => 2
[1,2,1,1,1,3] => 2
[1,2,1,1,2,1,1] => 2
[1,2,1,1,2,2] => 2
[1,2,1,1,3,1] => 2
[1,2,1,1,4] => 2
[1,2,1,2,1,1,1] => 2
[1,2,1,2,1,2] => 3
[1,2,1,2,2,1] => 2
[1,2,1,2,3] => 3
[1,2,1,3,1,1] => 2
[1,2,1,3,2] => 2
[1,2,1,4,1] => 2
[1,2,1,5] => 2
[1,2,2,1,1,1,1] => 1
[1,2,2,1,1,2] => 2
[1,2,2,1,2,1] => 2
[1,2,2,1,3] => 2
[1,2,2,2,1,1] => 1
[1,2,2,2,2] => 1
[1,2,2,3,1] => 2
[1,2,2,4] => 2
[1,2,3,1,1,1] => 2
[1,2,3,1,2] => 3
[1,2,3,2,1] => 2
[1,2,3,3] => 2
[1,2,4,1,1] => 2
[1,2,4,2] => 2
[1,2,5,1] => 2
[1,2,6] => 2
[1,3,1,1,1,1,1] => 1
[1,3,1,1,1,2] => 2
[1,3,1,1,2,1] => 2
[1,3,1,1,3] => 2
[1,3,1,2,1,1] => 2
[1,3,1,2,2] => 2
[1,3,1,3,1] => 2
[1,3,1,4] => 2
[1,3,2,1,1,1] => 1
[1,3,2,1,2] => 2
[1,3,2,2,1] => 1
[1,3,2,3] => 2
[1,3,3,1,1] => 1
[1,3,3,2] => 1
[1,3,4,1] => 2
[1,3,5] => 2
[1,4,1,1,1,1] => 1
[1,4,1,1,2] => 2
[1,4,1,2,1] => 2
[1,4,1,3] => 2
[1,4,2,1,1] => 1
[1,4,2,2] => 1
[1,4,3,1] => 1
[1,4,4] => 1
[1,5,1,1,1] => 1
[1,5,1,2] => 2
[1,5,2,1] => 1
[1,5,3] => 1
[1,6,1,1] => 1
[1,6,2] => 1
[1,7,1] => 1
[1,8] => 1
[2,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,2] => 1
[2,1,1,1,1,2,1] => 1
[2,1,1,1,1,3] => 1
[2,1,1,1,2,1,1] => 1
[2,1,1,1,2,2] => 1
[2,1,1,1,3,1] => 1
[2,1,1,1,4] => 1
[2,1,1,2,1,1,1] => 1
[2,1,1,2,1,2] => 2
[2,1,1,2,2,1] => 1
[2,1,1,2,3] => 2
[2,1,1,3,1,1] => 1
[2,1,1,3,2] => 1
[2,1,1,4,1] => 1
[2,1,1,5] => 1
[2,1,2,1,1,1,1] => 1
[2,1,2,1,1,2] => 2
[2,1,2,1,2,1] => 2
[2,1,2,1,3] => 2
[2,1,2,2,1,1] => 1
[2,1,2,2,2] => 1
[2,1,2,3,1] => 2
[2,1,2,4] => 2
[2,1,3,1,1,1] => 1
[2,1,3,1,2] => 2
[2,1,3,2,1] => 1
[2,1,3,3] => 1
[2,1,4,1,1] => 1
[2,1,4,2] => 1
[2,1,5,1] => 1
[2,1,6] => 1
[2,2,1,1,1,1,1] => 0
[2,2,1,1,1,2] => 1
[2,2,1,1,2,1] => 1
[2,2,1,1,3] => 1
[2,2,1,2,1,1] => 1
[2,2,1,2,2] => 1
[2,2,1,3,1] => 1
[2,2,1,4] => 1
[2,2,2,1,1,1] => 0
[2,2,2,1,2] => 1
[2,2,2,2,1] => 0
[2,2,2,3] => 1
[2,2,3,1,1] => 1
[2,2,3,2] => 1
[2,2,4,1] => 1
[2,2,5] => 1
[2,3,1,1,1,1] => 1
[2,3,1,1,2] => 2
[2,3,1,2,1] => 2
[2,3,1,3] => 2
[2,3,2,1,1] => 1
[2,3,2,2] => 1
[2,3,3,1] => 1
[2,3,4] => 2
[2,4,1,1,1] => 1
[2,4,1,2] => 2
[2,4,2,1] => 1
[2,4,3] => 1
[2,5,1,1] => 1
[2,5,2] => 1
[2,6,1] => 1
[2,7] => 1
[3,1,1,1,1,1,1] => 0
[3,1,1,1,1,2] => 1
[3,1,1,1,2,1] => 1
[3,1,1,1,3] => 1
[3,1,1,2,1,1] => 1
[3,1,1,2,2] => 1
[3,1,1,3,1] => 1
[3,1,1,4] => 1
[3,1,2,1,1,1] => 1
[3,1,2,1,2] => 2
[3,1,2,2,1] => 1
[3,1,2,3] => 2
[3,1,3,1,1] => 1
[3,1,3,2] => 1
[3,1,4,1] => 1
[3,1,5] => 1
[3,2,1,1,1,1] => 0
[3,2,1,1,2] => 1
[3,2,1,2,1] => 1
[3,2,1,3] => 1
[3,2,2,1,1] => 0
[3,2,2,2] => 0
[3,2,3,1] => 1
[3,2,4] => 1
[3,3,1,1,1] => 0
[3,3,1,2] => 1
[3,3,2,1] => 0
[3,3,3] => 0
[3,4,1,1] => 1
[3,4,2] => 1
[3,5,1] => 1
[3,6] => 1
[4,1,1,1,1,1] => 0
[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] => 1
[4,2,1,1,1] => 0
[4,2,1,2] => 1
[4,2,2,1] => 0
[4,2,3] => 1
[4,3,1,1] => 0
[4,3,2] => 0
[4,4,1] => 0
[4,5] => 1
[5,1,1,1,1] => 0
[5,1,1,2] => 1
[5,1,2,1] => 1
[5,1,3] => 1
[5,2,1,1] => 0
[5,2,2] => 0
[5,3,1] => 0
[5,4] => 0
[6,1,1,1] => 0
[6,1,2] => 1
[6,2,1] => 0
[6,3] => 0
[7,1,1] => 0
[7,2] => 0
[8,1] => 0
[9] => 0
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Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
References
[1] Carlitz, L. Enumeration of compositions by rises, falls and levels MathSciNet:0450078
Code
def statistic(pi):
    res = 0
    for i in range(len(pi)-1):
        if pi[i] < pi[i+1]:
            res += 1
    return res
Created
Apr 08, 2017 at 22:12 by Martin Rubey
Updated
Nov 06, 2017 at 23:38 by Martin Rubey