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Identifier

St000243: Permutations ⟶ ℤ

Values

[1,2] => 1

[2,1] => 1

[1,2,3] => 1

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

[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] => 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] => 1

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

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

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

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

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

[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] => 1

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

[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] => 1

[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] => 2

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

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

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

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

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

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

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

>>> Load all 897 entries. <<<

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Description

The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.

Code

def statistic(pi):
    pi = [pi[-1]] + list(pi) + [pi[0]]
    return sum( 1 for i in [1..len(pi)-2] if pi[i-1] > pi[i] < pi[i+1] )

Created

Feb 27, 2015 at 15:28 by Christian Stump

Updated

Apr 27, 2018 at 14:50 by Christian Stump