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Identifier

St000325: Permutations ⟶ ℤ (values match St000021The number of descents of a permutation., St000470The number of runs in a permutation.)

Values

[1] => 1

[1,2] => 1

[2,1] => 2

[1,2,3] => 1

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

[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] => 4

[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] => 2

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

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

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

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

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

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

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

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

[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] => 4

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The width 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 width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents St000021The number of descents of a permutation.. This also matches the number of runs in a permutation St000470The number of runs in a permutation..
See also St000308The height of the tree associated to a permutation. for the height of this tree.

References

[1] Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows. OEIS:A123125
[2] http://oeis.org/wiki/User:Peter_Luschny/PermutationTrees

Code

def statistic(pi):
    if pi == []: return 0
    w, i, branch, next = 1, 0, [0], pi[0]
    while true:
        while next < branch[len(branch)-1]:
            branch.pop()
        current = 0
        while next > current:
            i += 1
            if i == len(pi): return w
            branch.append(next)
            current, next = next, pi[i]
        w += 1

Created

Dec 11, 2015 at 11:05 by Peter Luschny

Updated

Mar 14, 2023 at 13:31 by Tilman Möller