Identifier
- St000092: Permutations ⟶ ℤ (values match St000353The number of inner valleys of a permutation.)
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 2
[1,3,2,5,4] => 2
[1,3,4,2,5] => 2
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 1
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 1
[1,5,2,3,4] => 2
[1,5,2,4,3] => 2
[1,5,3,2,4] => 2
[1,5,3,4,2] => 2
[1,5,4,2,3] => 2
[1,5,4,3,2] => 1
[2,1,3,4,5] => 2
[2,1,3,5,4] => 2
[2,1,4,3,5] => 3
[2,1,4,5,3] => 2
[2,1,5,3,4] => 3
[2,1,5,4,3] => 2
[2,3,1,4,5] => 2
[2,3,1,5,4] => 2
[2,3,4,1,5] => 2
[2,3,4,5,1] => 1
[2,3,5,1,4] => 2
[2,3,5,4,1] => 1
[2,4,1,3,5] => 2
[2,4,1,5,3] => 2
[2,4,3,1,5] => 2
[2,4,3,5,1] => 2
[2,4,5,1,3] => 2
[2,4,5,3,1] => 1
[2,5,1,3,4] => 2
[2,5,1,4,3] => 2
[2,5,3,1,4] => 2
[2,5,3,4,1] => 2
[2,5,4,1,3] => 2
[2,5,4,3,1] => 1
[3,1,2,4,5] => 2
[3,1,2,5,4] => 2
[3,1,4,2,5] => 3
[3,1,4,5,2] => 2
[3,1,5,2,4] => 3
[3,1,5,4,2] => 2
[3,2,1,4,5] => 2
[3,2,1,5,4] => 2
[3,2,4,1,5] => 3
[3,2,4,5,1] => 2
[3,2,5,1,4] => 3
[3,2,5,4,1] => 2
[3,4,1,2,5] => 2
[3,4,1,5,2] => 2
[3,4,2,1,5] => 2
[3,4,2,5,1] => 2
[3,4,5,1,2] => 2
[3,4,5,2,1] => 1
[3,5,1,2,4] => 2
[3,5,1,4,2] => 2
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Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
References
[1] Claesson, A., Kitaev, S. Classification of bijections between 321- and 132-avoiding permutations MathSciNet:2465405
Code
def statistic(pi):
pi = [0] + list(pi) + [0]
n = len(pi)
return sum( 1 for i in [1 .. n-2] if pi[i-1] < pi[i] > pi[i+1] )
Created
Jun 13, 2013 at 16:10 by Chris Berg
Updated
Jun 05, 2017 at 10:07 by Christian Stump
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