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Matching statistic: St001760
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St001760: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of prefix or suffix reversals needed to sort a permutation. A prefix reversal in a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of an initial subsequence of the form

$\operatorname{reversal}_{j}(\pi) = [\pi_j,\ldots,\pi_{1},\pi_{j+1},\ldots,\pi_n]$ for

$1 < j \leq n$ . Final reversals are defined analogously. This statistic is then given by the minimal number of initial or final reversals needed to sort a permutation.