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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 [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