- FindStat

edit this statistic or download as text // json

Identifier

St000216: Permutations ⟶ ℤ

Values

[1,2] => 0

[2,1] => 1

[1,2,3] => 0

[1,3,2] => 1

[2,1,3] => 1

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 872 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The absolute length of a permutation.
The absolute length of a permutation $\sigma$ of length $n$ is the shortest $k$ such that $\sigma = t_1 \dots t_k$ for transpositions $t_i$. Also, this is equal to $n$ minus the number of cycles of $\sigma$.

References

[1] Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind (1<=k<=n; in other words, the unsigned Stirling numbers of the first kind in reverse order). OEIS:A094638

Code

def statistic(pi):
      return len(pi) - len(pi.cycle_tuples())

Created

Aug 29, 2014 at 22:28 by Martin Rubey

Updated

Jul 03, 2019 at 19:02 by Christian Stump