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Identifier

St000060: Permutations ⟶ ℤ

Values

[1,2] => 1

[2,1] => 1

[1,2,3] => 2

[1,3,2] => 2

[2,1,3] => 1

[2,3,1] => 2

[3,1,2] => 1

[3,2,1] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 872 entries. <<<

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Description

The greater neighbor of the maximum.
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 smallest path leaf label of the binary tree associated to a permutation (St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation.), see also [3].

References

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

Code

def statistic(pi):
    n = pi.size()
    pi = [0]+list(pi)+[0]
    i = pi.index(n)
    return max(pi[i-1],pi[i+1])

Created

Apr 10, 2013 at 10:30 by Christian Stump

Updated

Mar 28, 2017 at 11:45 by Christian Stump