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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000308
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 1
[2,1] => [1,1,0,0]
=> [1,2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
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 [[St000325]] for the width of this tree.
Matching statistic: St000381
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 3
Description
The largest part of an integer composition.
Matching statistic: St000147
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1]
=> 1
[1,2] => [1,0,1,0]
=> [1,1] => [1,1]
=> 1
[2,1] => [1,1,0,0]
=> [2] => [2]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [3] => [3]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3] => [3]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000444
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001062
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> {{1}}
=> ? = 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
Description
The maximal size of a block of a set partition.
Matching statistic: St000392
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => => ? = 1 - 1
[1,2] => [1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [2,1] => 1 => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 10 => 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 010 => 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 100 => 1 = 2 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 100 => 1 = 2 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0010 => 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0100 => 1 = 2 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0100 => 1 = 2 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0101 => 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0110 => 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 2 = 3 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0110 => 2 = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000503
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> {{1}}
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> {{1},{2}}
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> {{1,2}}
=> 1 = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1 = 2 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1 = 2 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 2 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2 = 3 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1 = 2 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1 = 2 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2 = 3 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3 = 4 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 2 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 2 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2 = 3 - 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 3 - 1
Description
The maximal difference between two elements in a common block.
Matching statistic: St000844
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000844: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000844: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [.,.]
=> [1] => ? = 1
[1,2] => [2,1] => [[.,.],.]
=> [1,2] => 1
[2,1] => [1,2] => [.,[.,.]]
=> [2,1] => 2
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 2
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 3
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 3
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 2
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 2
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 3
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 4
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 4
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 2
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 2
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 2
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 3
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 2
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 2
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 2
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 3
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 3
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 3
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 3
[1,4,5,3,2] => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 3
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => ? = 3
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => ? = 3
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => ? = 4
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => ? = 4
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [[[[.,[[.,.],[.,.]]],.],.],.]
=> [2,4,3,1,5,6,7] => ? = 4
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [[[[.,[[.,.],[.,.]]],.],.],.]
=> [2,4,3,1,5,6,7] => ? = 4
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [[[[.,[.,[[.,.],.]]],.],.],.]
=> [3,4,2,1,5,6,7] => ? = 4
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => ? = 4
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3
[1,2,5,7,3,4,6] => [7,6,3,1,5,4,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3
[1,2,5,7,3,6,4] => [7,6,3,1,5,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3
[1,2,5,7,4,3,6] => [7,6,3,1,4,5,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3
[1,2,5,7,4,6,3] => [7,6,3,1,4,2,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3
[1,2,5,7,6,3,4] => [7,6,3,1,2,5,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3
[1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3
[1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4
[1,2,6,5,4,7,3] => [7,6,2,3,4,1,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4
[1,2,6,5,7,4,3] => [7,6,2,3,1,4,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4
[1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4
[1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => [[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => ? = 5
[1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => [[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => ? = 5
[1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => [[[.,[[[.,.],[.,.]],.]],.],.]
=> [2,4,3,5,1,6,7] => ? = 5
[1,2,7,3,5,6,4] => [7,6,1,5,3,2,4] => [[[.,[[[.,.],[.,.]],.]],.],.]
=> [2,4,3,5,1,6,7] => ? = 5
[1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => [[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => ? = 5
[1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => [[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => ? = 5
[1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5
[1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5
[1,2,7,4,5,3,6] => [7,6,1,4,3,5,2] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5
[1,2,7,4,5,6,3] => [7,6,1,4,3,2,5] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5
[1,2,7,4,6,3,5] => [7,6,1,4,2,5,3] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5
[1,2,7,4,6,5,3] => [7,6,1,4,2,3,5] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5
[1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5
[1,2,7,5,3,6,4] => [7,6,1,3,5,2,4] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5
[1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5
[1,2,7,5,4,6,3] => [7,6,1,3,4,2,5] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5
[1,2,7,5,6,3,4] => [7,6,1,3,2,5,4] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5
[1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5
[1,2,7,6,3,4,5] => [7,6,1,2,5,4,3] => [[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => ? = 5
[1,2,7,6,3,5,4] => [7,6,1,2,5,3,4] => [[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => ? = 5
[1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [3,5,4,2,1,6,7] => ? = 5
[1,2,7,6,4,5,3] => [7,6,1,2,4,3,5] => [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [3,5,4,2,1,6,7] => ? = 5
[1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => [[[.,[.,[.,[[.,.],.]]]],.],.]
=> [4,5,3,2,1,6,7] => ? = 5
[1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => ? = 5
[1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [[[[.,[[.,.],.]],.],[.,.]],.]
=> [2,3,1,4,6,5,7] => ? = 3
Description
The size of the largest block in the direct sum decomposition of a permutation.
A component of a permutation $\pi$ is a set of consecutive numbers $\{a,a+1,\dots, b\}$ such that $a\leq \pi(i) \leq b$ for all $a\leq i\leq b$.
This statistic is the size of the largest component which does not properly contain another component.
Matching statistic: St000956
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [.,.]
=> [1] => ? = 1 - 1
[1,2] => [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 2 - 1
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2 = 3 - 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 3 - 1
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 3 - 1
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 3 - 1
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 3 - 1
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 3 - 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 3 - 1
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3 = 4 - 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3 = 4 - 1
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3 = 4 - 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 2 = 3 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 2 = 3 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,5,3,2] => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => ? = 3 - 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => ? = 3 - 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => ? = 4 - 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => ? = 4 - 1
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [[[[.,[[.,.],[.,.]]],.],.],.]
=> [2,4,3,1,5,6,7] => ? = 4 - 1
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [[[[.,[[.,.],[.,.]]],.],.],.]
=> [2,4,3,1,5,6,7] => ? = 4 - 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [[[[.,[.,[[.,.],.]]],.],.],.]
=> [3,4,2,1,5,6,7] => ? = 4 - 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => ? = 4 - 1
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,5,7,3,4,6] => [7,6,3,1,5,4,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,7,3,6,4] => [7,6,3,1,5,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,7,4,3,6] => [7,6,3,1,4,5,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,7,4,6,3] => [7,6,3,1,4,2,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,7,6,3,4] => [7,6,3,1,2,5,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,5,4,7,3] => [7,6,2,3,4,1,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,5,7,4,3] => [7,6,2,3,1,4,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => [[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => ? = 5 - 1
[1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => [[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => ? = 5 - 1
[1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => [[[.,[[[.,.],[.,.]],.]],.],.]
=> [2,4,3,5,1,6,7] => ? = 5 - 1
[1,2,7,3,5,6,4] => [7,6,1,5,3,2,4] => [[[.,[[[.,.],[.,.]],.]],.],.]
=> [2,4,3,5,1,6,7] => ? = 5 - 1
[1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => [[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => ? = 5 - 1
[1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => [[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => ? = 5 - 1
[1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,5,3,6] => [7,6,1,4,3,5,2] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,5,6,3] => [7,6,1,4,3,2,5] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,6,3,5] => [7,6,1,4,2,5,3] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,6,5,3] => [7,6,1,4,2,3,5] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5 - 1
[1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5 - 1
[1,2,7,5,3,6,4] => [7,6,1,3,5,2,4] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5 - 1
[1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5 - 1
[1,2,7,5,4,6,3] => [7,6,1,3,4,2,5] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5 - 1
[1,2,7,5,6,3,4] => [7,6,1,3,2,5,4] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5 - 1
[1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5 - 1
[1,2,7,6,3,4,5] => [7,6,1,2,5,4,3] => [[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => ? = 5 - 1
[1,2,7,6,3,5,4] => [7,6,1,2,5,3,4] => [[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => ? = 5 - 1
[1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [3,5,4,2,1,6,7] => ? = 5 - 1
[1,2,7,6,4,5,3] => [7,6,1,2,4,3,5] => [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [3,5,4,2,1,6,7] => ? = 5 - 1
[1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => [[[.,[.,[.,[[.,.],.]]]],.],.]
=> [4,5,3,2,1,6,7] => ? = 5 - 1
[1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => ? = 5 - 1
[1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [[[[.,[[.,.],.]],.],[.,.]],.]
=> [2,3,1,4,6,5,7] => ? = 3 - 1
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St000209
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [.,.]
=> [1] => 0 = 1 - 1
[1,2] => [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1 = 2 - 1
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2 = 3 - 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 2 - 1
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2 = 3 - 1
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2 = 3 - 1
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 3 - 1
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 3 - 1
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 3 - 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2 = 3 - 1
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3 = 4 - 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 3 = 4 - 1
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 3 = 4 - 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1 = 2 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 1 = 2 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 2 = 3 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 2 = 3 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 1 = 2 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1 = 2 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 2 = 3 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => ? = 3 - 1
[1,2,3,4,7,6,5] => [7,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => ? = 3 - 1
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => ? = 4 - 1
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => [[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => ? = 4 - 1
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => [[[[.,[[.,.],[.,.]]],.],.],.]
=> [2,4,3,1,5,6,7] => ? = 4 - 1
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => [[[[.,[[.,.],[.,.]]],.],.],.]
=> [2,4,3,1,5,6,7] => ? = 4 - 1
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => [[[[.,[.,[[.,.],.]]],.],.],.]
=> [3,4,2,1,5,6,7] => ? = 4 - 1
[1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => ? = 4 - 1
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,5,7,3,4,6] => [7,6,3,1,5,4,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,7,3,6,4] => [7,6,3,1,5,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,7,4,3,6] => [7,6,3,1,4,5,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,7,4,6,3] => [7,6,3,1,4,2,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,7,6,3,4] => [7,6,3,1,2,5,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,7,6,4,3] => [7,6,3,1,2,4,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,5,4,7,3] => [7,6,2,3,4,1,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,5,7,4,3] => [7,6,2,3,1,4,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,6,7,5,4,3] => [7,6,2,1,3,4,5] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 4 - 1
[1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => [[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => ? = 5 - 1
[1,2,7,3,4,6,5] => [7,6,1,5,4,2,3] => [[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => ? = 5 - 1
[1,2,7,3,5,4,6] => [7,6,1,5,3,4,2] => [[[.,[[[.,.],[.,.]],.]],.],.]
=> [2,4,3,5,1,6,7] => ? = 5 - 1
[1,2,7,3,5,6,4] => [7,6,1,5,3,2,4] => [[[.,[[[.,.],[.,.]],.]],.],.]
=> [2,4,3,5,1,6,7] => ? = 5 - 1
[1,2,7,3,6,4,5] => [7,6,1,5,2,4,3] => [[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => ? = 5 - 1
[1,2,7,3,6,5,4] => [7,6,1,5,2,3,4] => [[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => ? = 5 - 1
[1,2,7,4,3,5,6] => [7,6,1,4,5,3,2] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,3,6,5] => [7,6,1,4,5,2,3] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,5,3,6] => [7,6,1,4,3,5,2] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,5,6,3] => [7,6,1,4,3,2,5] => [[[.,[[[.,.],.],[.,.]]],.],.]
=> [2,3,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,6,3,5] => [7,6,1,4,2,5,3] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5 - 1
[1,2,7,4,6,5,3] => [7,6,1,4,2,3,5] => [[[.,[[.,[.,.]],[.,.]]],.],.]
=> [3,2,5,4,1,6,7] => ? = 5 - 1
[1,2,7,5,3,4,6] => [7,6,1,3,5,4,2] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5 - 1
[1,2,7,5,3,6,4] => [7,6,1,3,5,2,4] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5 - 1
[1,2,7,5,4,3,6] => [7,6,1,3,4,5,2] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5 - 1
[1,2,7,5,4,6,3] => [7,6,1,3,4,2,5] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5 - 1
[1,2,7,5,6,3,4] => [7,6,1,3,2,5,4] => [[[.,[[.,.],[[.,.],.]]],.],.]
=> [2,4,5,3,1,6,7] => ? = 5 - 1
[1,2,7,5,6,4,3] => [7,6,1,3,2,4,5] => [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [2,5,4,3,1,6,7] => ? = 5 - 1
[1,2,7,6,3,4,5] => [7,6,1,2,5,4,3] => [[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => ? = 5 - 1
[1,2,7,6,3,5,4] => [7,6,1,2,5,3,4] => [[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => ? = 5 - 1
[1,2,7,6,4,3,5] => [7,6,1,2,4,5,3] => [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [3,5,4,2,1,6,7] => ? = 5 - 1
[1,2,7,6,4,5,3] => [7,6,1,2,4,3,5] => [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [3,5,4,2,1,6,7] => ? = 5 - 1
[1,2,7,6,5,3,4] => [7,6,1,2,3,5,4] => [[[.,[.,[.,[[.,.],.]]]],.],.]
=> [4,5,3,2,1,6,7] => ? = 5 - 1
[1,2,7,6,5,4,3] => [7,6,1,2,3,4,5] => [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => ? = 5 - 1
[1,3,4,7,2,5,6] => [7,5,4,1,6,3,2] => [[[[.,[[.,.],.]],.],[.,.]],.]
=> [2,3,1,4,6,5,7] => ? = 3 - 1
[1,3,4,7,2,6,5] => [7,5,4,1,6,2,3] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 3 - 1
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001372The length of a longest cyclic run of ones of a binary word. St001330The hat guessing number of a graph. St000141The maximum drop size of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000328The maximum number of child nodes in a tree. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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