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Identifier

St000724: Permutations ⟶ ℤ

Values

[1,2] => 2

[2,1] => 2

[1,2,3] => 3

[1,3,2] => 3

[2,1,3] => 2

[2,3,1] => 3

[3,1,2] => 2

[3,2,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2\ q^{2}$

$F_{3} = 2\ q^{2} + 4\ q^{3}$

$F_{4} = 4\ q^{2} + 8\ q^{3} + 12\ q^{4}$

$F_{5} = 12\ q^{2} + 24\ q^{3} + 36\ q^{4} + 48\ q^{5}$

$F_{6} = 48\ q^{2} + 96\ q^{3} + 144\ q^{4} + 192\ q^{5} + 240\ q^{6}$

Description

The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.
Associate an increasing binary tree to the permutation using Mp00061to increasing tree. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1].
Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations

$\pi$ such that

$\pi(1) < \pi(2) > \pi(3) \cdots$ ) with the greater neighbor of the maximum (St000060The greater neighbor of the maximum.), see also [3].

References

[1] Poupard, C. Deux propriétés des arbres binaires ordonnés stricts MathSciNet:1005843
[2] https://www.emis.de/journals/SLC/wpapers/s74vortrag/han.pdf
[3] Foata, D., Han, G.-N. Finite difference calculus for alternating permutations MathSciNet:3173528 arXiv:1304.2483

Code

def statistic(pi):
    pi = list(pi)
    if len(pi) == 1:
        return pi[0]
    else:
        a = min(pi)
        j = pi.index(a)
        b = min( x for x in pi if x != a )
        if pi.index(b) < j:
            return statistic(pi[:j])
        else:
            return statistic(pi[j+1:])

Created

Mar 28, 2017 at 11:20 by Christian Stump

Updated

Mar 28, 2017 at 11:47 by Christian Stump