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Identifier

St000308: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 2

[2,1] => 1

[1,2,3] => 3

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also St000325The width of the tree associated to a permutation. for the width of this tree.

References

[1] Permutation trees of power n and height k. OEIS:A179454
[2] http://oeis.org/wiki/User:Peter_Luschny/PermutationTrees

Code

def statistic(pi):
    if pi == []: return 0
    
    root, i, h = 0, 0, 0 
    branch, next = [root], pi[0]

    while true:
        while next < branch[len(branch)-1]:
            del(branch[len(branch)-1])
        current = root

        while next > current:
            i += 1
            branch.append(next)
            h = max(h, len(branch))
            if i >= len(pi): return h-1
            current, next = next, pi[i]

Created

Dec 08, 2015 at 08:43 by Peter Luschny

Updated

Feb 25, 2022 at 16:58 by Nadia Lafreniere