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Identifier

St000099: Permutations ⟶ ℤ (values match St000023The number of inner peaks of a permutation.)

Values

[1] => 1

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 2

[2,1,3] => 1

[2,3,1] => 2

[3,1,2] => 1

[3,2,1] => 1

[1,2,3,4] => 1

[1,2,4,3] => 2

[1,3,2,4] => 2

[1,3,4,2] => 2

[1,4,2,3] => 2

[1,4,3,2] => 2

[2,1,3,4] => 1

[2,1,4,3] => 2

[2,3,1,4] => 2

[2,3,4,1] => 2

[2,4,1,3] => 2

[2,4,3,1] => 2

[3,1,2,4] => 1

[3,1,4,2] => 2

[3,2,1,4] => 1

[3,2,4,1] => 2

[3,4,1,2] => 2

[3,4,2,1] => 2

[4,1,2,3] => 1

[4,1,3,2] => 2

[4,2,1,3] => 1

[4,2,3,1] => 2

[4,3,1,2] => 1

[4,3,2,1] => 1

[1,2,3,4,5] => 1

[1,2,3,5,4] => 2

[1,2,4,3,5] => 2

[1,2,4,5,3] => 2

[1,2,5,3,4] => 2

[1,2,5,4,3] => 2

[1,3,2,4,5] => 2

[1,3,2,5,4] => 3

[1,3,4,2,5] => 2

[1,3,4,5,2] => 2

[1,3,5,2,4] => 2

[1,3,5,4,2] => 2

[1,4,2,3,5] => 2

[1,4,2,5,3] => 3

[1,4,3,2,5] => 2

[1,4,3,5,2] => 3

[1,4,5,2,3] => 2

[1,4,5,3,2] => 2

[1,5,2,3,4] => 2

[1,5,2,4,3] => 3

[1,5,3,2,4] => 2

[1,5,3,4,2] => 3

[1,5,4,2,3] => 2

[1,5,4,3,2] => 2

[2,1,3,4,5] => 1

[2,1,3,5,4] => 2

[2,1,4,3,5] => 2

[2,1,4,5,3] => 2

[2,1,5,3,4] => 2

[2,1,5,4,3] => 2

[2,3,1,4,5] => 2

[2,3,1,5,4] => 3

[2,3,4,1,5] => 2

[2,3,4,5,1] => 2

[2,3,5,1,4] => 2

[2,3,5,4,1] => 2

[2,4,1,3,5] => 2

[2,4,1,5,3] => 3

[2,4,3,1,5] => 2

[2,4,3,5,1] => 3

[2,4,5,1,3] => 2

[2,4,5,3,1] => 2

[2,5,1,3,4] => 2

[2,5,1,4,3] => 3

[2,5,3,1,4] => 2

[2,5,3,4,1] => 3

[2,5,4,1,3] => 2

[2,5,4,3,1] => 2

[3,1,2,4,5] => 1

[3,1,2,5,4] => 2

[3,1,4,2,5] => 2

[3,1,4,5,2] => 2

[3,1,5,2,4] => 2

[3,1,5,4,2] => 2

[3,2,1,4,5] => 1

[3,2,1,5,4] => 2

[3,2,4,1,5] => 2

[3,2,4,5,1] => 2

[3,2,5,1,4] => 2

[3,2,5,4,1] => 2

[3,4,1,2,5] => 2

[3,4,1,5,2] => 3

[3,4,2,1,5] => 2

[3,4,2,5,1] => 3

[3,4,5,1,2] => 2

[3,4,5,2,1] => 2

[3,5,1,2,4] => 2

[3,5,1,4,2] => 3

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Description

The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is St000353The number of inner valleys of a permutation..

References

[1] Claesson, A., Kitaev, S. Classification of bijections between 321- and 132-avoiding permutations MathSciNet:2465405

Code

def statistic(pi):
    pi1 = [len(pi)+1] + list(pi) + [len(pi)+1]
    return len([i for i in range(1,len(pi1)-1) if pi1[i-1] > pi1[i] < pi1[i+1]])

Created

Jun 13, 2013 at 16:16 by Chris Berg

Updated

Jan 08, 2016 at 14:15 by Martin Rubey