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Identifier

St000236: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 2

[2,1] => 2

[1,2,3] => 3

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 3

[3,1,2] => 0

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \in \{ i,i+1 \}$ considered cyclically.

References

[1] Li, Y. Ménage Numbers and Ménage Permutations arXiv:1502.06068

Code

def statistic(pi):
    n = len(pi)
    return sum( 1 for i in range(n) if pi[i] in [ i+1, (i+1) % n + 1 ] )

Created

Feb 24, 2015 at 13:19 by Christian Stump

Updated

May 17, 2023 at 00:17 by Will Dowling