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Identifier

St000162: Permutations ⟶ ℤ

Values

[1] => 0

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 1

[3,1,2] => 1

[3,2,1] => 1

[1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

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Description

The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see St000031The number of cycles in the cycle decomposition of a permutation.) and the number of fixed points of $\pi$ (see St000022The number of fixed points of a permutation.).

References

[1] Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)). OEIS:A136394

Code

def statistic(pi):
    return sum( 1 for i in pi.cycle_type() if i > 1 )

Created

Sep 21, 2013 at 17:17 by Christian Stump

Updated

Dec 30, 2019 at 11:38 by Martin Rubey