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 $a_i < a_{i+1}$.
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
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