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Identifier

St000670: 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] => 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] => 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] => 2

[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] => 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] => 2

[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] => 2

[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] => 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] => 2

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

>>> Load all 1770 entries. <<<

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Description

The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.

Code

def children(pi):
    pi = list(pi)
    n = len(pi)
    for j in range(n):
        for i in range(j):
            yield Permutation(pi[:i]+list(reversed(pi[i:j+1]))+pi[j+1:])

@cached_function
def statistic_classes(n):
    A = RecursivelyEnumeratedSet([Permutation([1..n])], children,structure='symmetric')
    return list(A.graded_component_iterator())

def statistic(pi):
    comps = statistic_classes(len(pi))
    for rank,comp in enumerate(comps):
        if pi in comp:
            return rank

Created

Dec 15, 2016 at 17:57 by Christian Stump

Updated

Jun 21, 2018 at 09:53 by Martin Rubey