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Your data matches 21 different statistics following compositions of up to 3 maps.
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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
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [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
[1,2,3,5,4] => [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]
=> 2
[1,2,4,5,3] => [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]
=> 3
[1,2,5,4,3] => [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]
=> 2
[1,3,2,5,4] => [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]
=> 2
[1,3,4,5,2] => [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]
=> 2
[1,3,5,4,2] => [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]
=> 3
[1,4,2,5,3] => [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]
=> 3
[1,4,3,5,2] => [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]
=> 3
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The length of the maximal rise of a Dyck path.
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
[1,2] => [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
[1,4,5,3,2] => [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
[1,2] => [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
[1,4,5,3,2] => [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: 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: 94% ●values known / values provided: 94%●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: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[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
[1,3,6,7,5,8,4,2] => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,5,8,4,2] => ? => ? = 3 - 1
[1,4,5,3,7,8,6,2] => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,3,7,8,6,2] => ? => ? = 3 - 1
[1,4,3,5,6,7,2,8] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,4,3,5,6,7,2,8] => ? => ? = 3 - 1
[1,2,4,6,5,7,3,8] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,2,4,6,5,7,3,8] => ? => ? = 2 - 1
[1,3,2,4,6,7,5,8] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,2,4,6,7,5,8] => ? => ? = 2 - 1
[1,4,3,5,2,6,7,8] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,3,5,2,6,7,8] => ? => ? = 3 - 1
[1,2,4,5,3,6,7,8] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,2,4,5,3,6,7,8] => ? => ? = 2 - 1
[1,3,6,4,5,7,8,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ? => ? = 3 - 1
[1,3,7,4,5,6,8,2] => [1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,7,6,5,4,8,2] => ? => ? = 4 - 1
[1,5,3,4,6,2,8,7] => [1,0,1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,5,4,3,6,2,8,7] => ? => ? = 4 - 1
[1,5,3,4,7,6,2,8] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,5,4,3,7,6,2,8] => ? => ? = 4 - 1
[2,1,3,5,8,6,7,4] => [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,3,5,8,7,6,4] => ? => ? = 3 - 1
[2,1,6,4,5,7,8,3] => [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,6,5,4,7,8,3] => ? => ? = 4 - 1
[2,1,4,6,3,7,5,8] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,5,7,3,8] => ? => ? = 2 - 1
[2,1,4,6,3,7,8,5] => [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,1,4,6,5,7,8,3] => ? => ? = 2 - 1
[2,1,4,5,7,3,8,6] => [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,4,5,7,6,8,3] => ? => ? = 2 - 1
[2,1,5,6,3,7,4,8] => [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,1,5,6,4,7,3,8] => ? => ? = 3 - 1
[2,1,5,7,3,4,6,8] => [1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,5,7,6,4,3,8] => ? => ? = 3 - 1
[2,1,3,5,8,4,6,7] => [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,3,5,8,7,6,4] => ? => ? = 3 - 1
[2,1,3,5,6,8,4,7] => [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,3,5,6,8,7,4] => ? => ? = 2 - 1
[2,1,6,3,7,4,5,8] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,5,7,4,3,8] => ? => ? = 4 - 1
[2,1,6,3,4,7,8,5] => [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,6,5,4,7,8,3] => ? => ? = 4 - 1
[2,1,3,6,7,4,8,5] => [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,3,6,7,5,8,4] => ? => ? = 3 - 1
[2,1,3,6,4,5,7,8] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,5,4,7,8] => ? => ? = 3 - 1
[2,1,3,7,4,8,5,6] => [1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,6,8,5,4] => ? => ? = 4 - 1
[2,1,3,4,7,5,6,8] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ? => ? = 3 - 1
[2,3,1,4,7,5,6,8] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,7,6,5,8] => ? => ? = 3 - 1
[2,3,1,4,5,8,6,7] => [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5,8,7,6] => ? => ? = 3 - 1
[1,3,5,2,4,8,6,7] => [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,3,5,4,2,8,7,6] => ? => ? = 3 - 1
[1,3,5,2,6,4,8,7] => [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,3,5,4,6,2,8,7] => ? => ? = 2 - 1
[1,3,5,6,2,4,8,7] => [1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,3,5,6,4,2,8,7] => ? => ? = 2 - 1
[1,3,5,2,6,8,4,7] => [1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,5,4,6,8,7,2] => ? => ? = 2 - 1
[1,3,5,6,2,8,4,7] => [1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,3,5,6,4,8,7,2] => ? => ? = 2 - 1
[1,3,6,2,4,7,8,5] => [1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,3,6,5,4,7,8,2] => ? => ? = 3 - 1
[1,3,6,7,2,4,8,5] => [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,6,7,5,4,8,2] => ? => ? = 3 - 1
[1,3,4,6,2,5,8,7] => [1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,4,6,5,2,8,7] => ? => ? = 2 - 1
[1,3,4,6,8,2,5,7] => [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,8,7,5,2] => ? => ? = 2 - 1
[1,3,6,2,4,5,7,8] => [1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,3,6,5,4,2,7,8] => ? => ? = 3 - 1
[1,3,4,6,2,5,7,8] => [1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,4,6,5,2,7,8] => ? => ? = 2 - 1
[1,3,7,2,4,8,5,6] => [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,7,6,5,8,4,2] => ? => ? = 4 - 1
[1,3,7,2,8,4,5,6] => [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,7,6,8,5,4,2] => ? => ? = 4 - 1
[1,3,7,2,4,5,8,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,7,6,5,4,8,2] => ? => ? = 4 - 1
[1,3,2,4,7,8,5,6] => [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,2,4,7,8,6,5] => ? => ? = 3 - 1
[1,3,4,7,8,2,5,6] => [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,7,8,6,5,2] => ? => ? = 3 - 1
[1,3,4,7,2,5,6,8] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,4,7,6,5,2,8] => ? => ? = 3 - 1
[1,4,5,2,7,3,6,8] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,4,5,3,7,6,2,8] => ? => ? = 3 - 1
[1,4,5,7,2,3,6,8] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,4,5,7,6,3,2,8] => ? => ? = 3 - 1
[1,4,2,5,7,3,8,6] => [1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,4,3,5,7,6,8,2] => ? => ? = 3 - 1
[1,4,5,2,7,3,8,6] => [1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,4,5,3,7,6,8,2] => ? => ? = 3 - 1
[1,4,5,7,2,3,8,6] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,4,5,7,6,3,8,2] => ? => ? = 3 - 1
Description
The length of the longest run of ones in a binary word.
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: 65% ●values known / values provided: 65%●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: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[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
[2,3,1,4,5,6,7,8] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 2
[2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 2
[1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,7,8,6,5,4,3,2] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[1,6,8,7,5,4,3,2] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5,6},{8}}
=> ? = 5
[1,7,6,8,5,4,3,2] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,6,7},{5},{8}}
=> ? = 6
[1,6,7,8,5,4,3,2] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5},{6},{8}}
=> ? = 5
[1,5,8,7,6,4,3,2] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4,5,6},{8}}
=> ? = 4
[1,6,5,8,7,4,3,2] => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4,5},{8}}
=> ? = 5
[1,7,6,5,8,4,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,5,6,7},{4},{8}}
=> ? = 6
[1,6,7,5,8,4,3,2] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 5
[1,5,7,6,8,4,3,2] => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4},{5,6},{8}}
=> ? = 4
[1,6,5,7,8,4,3,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,6,7},{4},{5},{8}}
=> ? = 5
[1,5,6,7,8,4,3,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4},{5},{6},{8}}
=> ? = 4
[1,4,8,7,6,5,3,2] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4,5,6},{8}}
=> ? = 4
[1,4,7,8,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4,6},{5},{8}}
=> ? = 3
[1,4,6,8,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,6},{4,5},{8}}
=> ? = 3
[1,4,6,7,8,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,6},{4},{5},{8}}
=> ? = 3
[1,5,4,7,8,6,3,2] => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
=> {{1,2,6,7},{3,5},{4},{8}}
=> ? = 4
[1,4,5,8,7,6,3,2] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4,5},{6},{8}}
=> ? = 3
[1,4,5,7,8,6,3,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> {{1,2,7},{3,5},{4},{6},{8}}
=> ? = 3
[1,6,5,4,8,7,3,2] => [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,5,6,7},{3,4},{8}}
=> ? = 5
[1,4,6,5,8,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4},{5,6},{8}}
=> ? = 3
[1,5,4,6,8,7,3,2] => [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,6,7},{3,4},{5},{8}}
=> ? = 4
[1,7,6,5,4,8,3,2] => [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 6
[1,4,7,6,5,8,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3},{4,5,6},{8}}
=> ? = 3
[1,4,6,7,5,8,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3},{4,6},{5},{8}}
=> ? = 3
[1,6,5,4,7,8,3,2] => [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,5,6,7},{3},{4},{8}}
=> ? = 5
[1,5,4,6,7,8,3,2] => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,6,7},{3},{4},{5},{8}}
=> ? = 4
[1,4,5,6,7,8,3,2] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ? = 3
[1,3,8,7,6,5,4,2] => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,4,5,6},{8}}
=> ? = 5
[1,3,7,8,6,5,4,2] => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,4,6},{5},{8}}
=> ? = 4
[1,3,6,8,7,5,4,2] => [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,6},{4,5},{8}}
=> ? = 3
[1,3,6,7,8,5,4,2] => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,3,6},{4},{5},{8}}
=> ? = 3
[1,3,5,8,7,6,4,2] => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3,4,5},{8}}
=> ? = 3
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 2
[1,3,5,6,8,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3,4},{5},{8}}
=> ? = 2
[1,3,6,7,5,8,4,2] => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,4,6},{3},{5},{8}}
=> ? = 3
[1,3,5,7,6,8,4,2] => [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3},{4,5},{8}}
=> ? = 2
[1,3,6,5,7,8,4,2] => [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> {{1,7},{2,5,6},{3},{4},{8}}
=> ? = 3
[1,3,5,6,7,8,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3},{4},{5},{8}}
=> ? = 2
[1,4,3,8,7,6,5,2] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,3,4,5},{8}}
=> ? = 4
[1,4,3,7,8,6,5,2] => [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,3,5},{4},{8}}
=> ? = 3
[1,4,3,6,8,7,5,2] => [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,5},{3,4},{8}}
=> ? = 3
[1,4,3,7,6,8,5,2] => [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,4,5},{3},{8}}
=> ? = 3
[1,4,3,6,7,8,5,2] => [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> {{1,6,7},{2,5},{3},{4},{8}}
=> ? = 3
[1,3,4,8,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> {{1,7},{2,3,4,5},{6},{8}}
=> ? = 4
[1,3,4,7,6,8,5,2] => [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0,1,0]
=> {{1,7},{2,4,5},{3},{6},{8}}
=> ? = 3
[1,3,4,6,7,8,5,2] => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> {{1,7},{2,5},{3},{4},{6},{8}}
=> ? = 2
[1,5,4,3,8,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> {{1,5,6,7},{2,3,4},{8}}
=> ? = 4
Description
The maximal size of a block of a set partition.
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: 58% ●values known / values provided: 58%●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: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[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
[2,3,1,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2 - 1
[2,1,3,4,5,6,7,8] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2 - 1
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,6,8,7,5,4,3,2] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 5 - 1
[1,6,7,8,5,4,3,2] => [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 5 - 1
[1,5,8,7,6,4,3,2] => [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 4 - 1
[1,6,5,8,7,4,3,2] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 5 - 1
[1,6,7,5,8,4,3,2] => [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 5 - 1
[1,5,7,6,8,4,3,2] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> {{1},{2,3,4,5},{6,7},{8}}
=> ? = 4 - 1
[1,6,5,7,8,4,3,2] => [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 5 - 1
[1,5,6,7,8,4,3,2] => [.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 4 - 1
[1,4,8,7,6,5,3,2] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[1,4,7,8,6,5,3,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 3 - 1
[1,4,6,8,7,5,3,2] => [.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6},{7,8}}
=> ? = 3 - 1
[1,4,6,7,8,5,3,2] => [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3,4},{5,6},{7},{8}}
=> ? = 3 - 1
[1,5,4,7,8,6,3,2] => [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> {{1},{2,3,4,5},{6,7},{8}}
=> ? = 4 - 1
[1,4,5,8,7,6,3,2] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 3 - 1
[1,4,5,7,8,6,3,2] => ?
=> ?
=> ?
=> ? = 3 - 1
[1,6,5,4,8,7,3,2] => ?
=> ?
=> ?
=> ? = 5 - 1
[1,4,6,5,8,7,3,2] => [.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6},{7,8}}
=> ? = 3 - 1
[1,5,4,6,8,7,3,2] => ?
=> ?
=> ?
=> ? = 4 - 1
[1,4,7,6,5,8,3,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 3 - 1
[1,4,6,7,5,8,3,2] => [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3,4},{5,6},{7},{8}}
=> ? = 3 - 1
[1,6,5,4,7,8,3,2] => [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> {{1},{2,3,4,5,6},{7},{8}}
=> ? = 5 - 1
[1,5,4,6,7,8,3,2] => [.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 4 - 1
[1,4,5,6,7,8,3,2] => [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 3 - 1
[1,3,8,7,6,5,4,2] => [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 5 - 1
[1,3,7,8,6,5,4,2] => [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3},{4,5,6,7},{8}}
=> ? = 4 - 1
[1,3,6,8,7,5,4,2] => [.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3},{4,5,6},{7,8}}
=> ? = 3 - 1
[1,3,6,7,8,5,4,2] => [.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 3 - 1
[1,3,5,8,7,6,4,2] => [.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5},{6,7,8}}
=> ? = 3 - 1
[1,3,5,7,8,6,4,2] => [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 2 - 1
[1,3,5,6,8,7,4,2] => [.,[[.,.],[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2,3},{4,5},{6},{7,8}}
=> ? = 2 - 1
[1,3,6,7,5,8,4,2] => [.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 3 - 1
[1,3,5,7,6,8,4,2] => ?
=> ?
=> ?
=> ? = 2 - 1
[1,3,6,5,7,8,4,2] => [.,[[.,.],[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3},{4,5,6},{7},{8}}
=> ? = 3 - 1
[1,3,5,6,7,8,4,2] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 2 - 1
[1,4,3,8,7,6,5,2] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4},{5,6,7,8}}
=> ? = 4 - 1
[1,4,3,7,8,6,5,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 3 - 1
[1,4,3,6,8,7,5,2] => [.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6},{7,8}}
=> ? = 3 - 1
[1,4,3,7,6,8,5,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 3 - 1
[1,4,3,6,7,8,5,2] => [.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3,4},{5,6},{7},{8}}
=> ? = 3 - 1
[1,3,4,8,7,6,5,2] => [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3},{4},{5,6,7,8}}
=> ? = 4 - 1
[1,3,4,7,6,8,5,2] => [.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3},{4},{5,6,7},{8}}
=> ? = 3 - 1
[1,3,4,6,7,8,5,2] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5,6},{7},{8}}
=> ? = 2 - 1
[1,5,4,3,8,7,6,2] => [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> {{1},{2,3,4,5},{6,7,8}}
=> ? = 4 - 1
[1,4,5,3,8,7,6,2] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 3 - 1
[1,4,5,3,7,8,6,2] => [.,[[[.,.],.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3,4},{5},{6,7},{8}}
=> ? = 3 - 1
[1,3,5,4,8,7,6,2] => [.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1},{2,3},{4,5},{6,7,8}}
=> ? = 3 - 1
[1,3,5,4,7,8,6,2] => ?
=> ?
=> ?
=> ? = 2 - 1
Description
The maximal difference between two elements in a common block.
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: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[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
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
[1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 2
[1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 2
[1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 2
[1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? = 2
[1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 2
[1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 2
[1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 2
[1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 2
[1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 2
[1,2,4,3,5,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? = 2
[1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? = 2
[1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => ? = 2
[1,2,4,3,7,5,6] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 3
[1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 3
[1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => ? = 2
[1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => ? = 2
[1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => ? = 2
[1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 2
[1,2,4,5,7,3,6] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 2
[1,2,4,5,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 2
[1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => ? = 2
[1,2,4,6,3,7,5] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => ? = 2
[1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,6,2,1] => ? = 2
[1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,7,2,1] => ? = 2
[1,2,4,6,7,3,5] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => ? = 2
[1,2,4,6,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,7,2,1] => ? = 2
[1,2,5,3,4,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 3
[1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 3
[1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 3
[1,2,5,4,3,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 3
[1,2,5,4,7,3,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 3
[1,2,5,4,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => ? = 3
[1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? = 2
[1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => ? = 2
[1,3,2,4,6,5,7] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => ? = 2
[1,3,2,4,6,7,5] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => ? = 2
[1,3,2,4,7,5,6] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => ? = 3
[1,3,2,4,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => ? = 3
[1,3,2,5,4,6,7] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => ? = 2
[1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => ? = 2
[1,3,2,5,6,4,7] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => ? = 2
[1,3,2,5,6,7,4] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => ? = 2
[1,3,2,5,7,4,6] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => ? = 2
[1,3,2,5,7,6,4] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => ? = 2
[1,3,2,6,4,5,7] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => ? = 3
[1,3,2,6,4,7,5] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => ? = 3
[1,3,2,6,5,4,7] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => ? = 3
[1,3,2,6,5,7,4] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => ? = 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: 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: 31% ●values known / values provided: 31%●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: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[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
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => ? = 2
[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
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[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
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2
[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
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3
[1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3
[1,2,5,4,7,3,6] => [7,6,3,4,1,5,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3
[1,2,5,4,7,6,3] => [7,6,3,4,1,2,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 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
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: 31% ●values known / values provided: 31%●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: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[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
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => ? = 2 - 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
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2 - 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
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 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
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,4,7,3,6] => [7,6,3,4,1,5,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,4,7,6,3] => [7,6,3,4,1,2,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 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
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: 31% ●values known / values provided: 31%●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: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[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
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => ? = 2 - 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
[1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2 - 1
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2 - 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
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [[[[.,[[.,.],.]],[.,.]],.],.]
=> [2,3,1,5,4,6,7] => ? = 3 - 1
[1,2,4,3,7,6,5] => [7,6,4,5,1,2,3] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 3 - 1
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 1
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 2 - 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
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [2,1,4,5,3,6,7] => ? = 3 - 1
[1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,4,7,3,6] => [7,6,3,4,1,5,2] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 3 - 1
[1,2,5,4,7,6,3] => [7,6,3,4,1,2,5] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 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
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. St000141The maximum drop size of a permutation. St001330The hat guessing number of a graph. 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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