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Identifier

St001579: 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] => 3

[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] => 4

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

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

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

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

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

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

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

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

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

[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] => 3

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

[1,5,4,3,2] => 6

[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] => 4

[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] => 5

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

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

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

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

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

[2,4,5,3,1] => 6

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

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

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

[2,5,3,4,1] => 6

[2,5,4,1,3] => 6

[2,5,4,3,1] => 7

[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] => 5

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

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

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

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

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

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

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

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

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

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

[3,4,5,1,2] => 6

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

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

[3,5,1,4,2] => 6

>>> Load all 1200 entries. <<<

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Description

The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation.
This is for a permutation $\sigma$ of length $n$ and the set $T = \{ (1,2), \dots, (n-1,n), (1,n) \}$ given by
$$\min\{ k \mid \sigma = t_1\dots t_k \text{ for } t_i \in T \text{ such that } t_1\dots t_j \text{ has more cyclic descents than } t_1\dots t_{j-1} \text{ for all } j\}.$$

References

[1] Winter, M. A graph similar to the Bruhat graph, what is it called? MathOverflow:366504

Code

def swap(pi,i):
    n = len(pi)
    if i > 0:
        if pi[i] < pi[i-1]:
            return pi[:i-1] + tuple([pi[i],pi[i-1]]) + pi[i+1:]
    else:
        if pi[0] > pi[-1]:
            return (pi[-1],) + pi[1:-1] + (pi[0],)

@cached_function
def statistic(pi):
    if pi is None:
        return infinity
    pi = tuple(pi)
    n = len(pi)
    if pi == tuple([1 .. n]):
        return 0
    return min( statistic(swap(pi,i)) for i in range(len(pi)) ) + 1

Created

Jul 26, 2020 at 11:22 by Christian Stump

Updated

Jan 22, 2024 at 15:09 by Martin Rubey