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Your data matches 78 different statistics following compositions of up to 3 maps.
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Matching statistic: St000141
(load all 17 compositions to match this statistic)

Mapping

St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000147

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 92%

Values

[1] => [1,0]
 => [1] => [1]
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,2] => [1,1]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [2,1] => [2]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => [3]
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,1,1]
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [3,1]
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,1,1]
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,1,1]
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,1]
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [3,1]
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => 3 = 2 + 1
[4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [6,3,4,5,2,1,7,8] => ?
 => ? = 3 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,8,4,5,7,6,1] => ?
 => ? = 3 + 1
[4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ?
 => ? = 3 + 1
[3,2,6,5,7,4,8,1] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ?
 => ? = 3 + 1
[4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ?
 => ? = 3 + 1
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,8,4,6,7,5,2] => ?
 => ? = 3 + 1
[2,1,6,5,7,4,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ?
 => ? = 3 + 1
[2,4,5,6,7,1,8,3] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,7,3,4,5,6,8,1] => ?
 => ? = 2 + 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,8,4,5,7,6,3,1] => ?
 => ? = 4 + 1
[2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,8,4,7,5,6,3,1] => ?
 => ? = 4 + 1
[2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ?
 => ? = 3 + 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] => ?
 => ? = 3 + 1
[2,6,1,3,4,5,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,6,5,4,3,1,7,8] => ?
 => ? = 4 + 1
[3,1,5,6,2,8,4,7] => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => [3,2,6,4,5,8,7,1] => ?
 => ? = 2 + 1
[1,3,5,8,2,4,6,7] => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,3,8,4,7,6,5,2] => ?
 => ? = 4 + 1
[3,1,6,2,7,4,8,5] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ?
 => ? = 3 + 1
[3,6,1,7,8,2,4,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => [8,2,5,4,6,7,3,1] => ?
 => ? = 4 + 1
[4,1,5,6,8,2,3,7] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => [8,3,2,4,5,7,6,1] => ?
 => ? = 3 + 1
[4,1,5,2,7,3,8,6] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ?
 => ? = 3 + 1
[4,1,5,2,7,8,3,6] => [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
 => [5,3,2,4,8,6,7,1] => ?
 => ? = 3 + 1
[4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
 => [8,3,4,6,5,2,7,1] => ?
 => ? = 4 + 1
[4,5,1,7,8,2,3,6] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => [8,3,4,2,6,7,5,1] => ?
 => ? = 3 + 1
[4,5,7,8,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [8,3,4,7,5,6,2,1] => ?
 => ? = 4 + 1
[1,4,2,5,3,6,7,8] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [1,4,3,5,2,6,7,8] => ?
 => ? = 2 + 1
[1,4,5,6,7,2,3,8] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,7,3,4,5,6,2,8] => ?
 => ? = 2 + 1
[4,1,6,2,7,3,8,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ?
 => ? = 3 + 1
[4,7,1,2,3,5,6,8] => ?
 => ? => ?
 => ? = 5 + 1
[1,4,2,7,3,5,6,8] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,3,7,6,5,2,8] => ?
 => ? = 3 + 1
[1,5,2,6,3,7,4,8] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,6,4,3,5,7,2,8] => ?
 => ? = 3 + 1
[1,5,2,7,3,4,6,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,7,4,3,6,5,2,8] => ?
 => ? = 3 + 1
[1,5,2,8,3,4,6,7] => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,8,4,3,7,6,5,2] => ?
 => ? = 4 + 1
[1,5,8,2,3,4,6,7] => ?
 => ? => ?
 => ? = 5 + 1
[1,6,2,7,3,4,5,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,7,5,4,6,3,2,8] => ?
 => ? = 4 + 1
[1,6,2,3,4,7,5,8] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,6,5,4,3,7,2,8] => ?
 => ? = 4 + 1
[5,4,7,2,1,8,3,6] => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [8,4,3,6,5,2,7,1] => ?
 => ? = 4 + 1
[3,6,1,8,7,2,5,4] => [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
 => [8,2,5,4,7,6,3,1] => ?
 => ? = 4 + 1
[6,3,2,5,4,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? => ?
 => ? = 5 + 1
[2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? => ?
 => ? = 3 + 1
[6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? => ?
 => ? = 5 + 1
[8,9,5,6,3,4,10,1,2,7] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0,0,0]
 => [10,8,7,5,6,4,3,9,2,1] => ?
 => ? = 7 + 1
[2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? => ?
 => ? = 3 + 1
[2,1,4,3,10,7,6,9,8,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? => ?
 => ? = 5 + 1
[6,7,8,9,10,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [10,9,4,5,6,7,8,3,2,1] => ?
 => ? = 5 + 1
[7,3,2,8,9,10,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0,0,0]
 => ? => ?
 => ? = 6 + 1
[8,5,4,3,2,9,10,1,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0,0,0]
 => ? => ?
 => ? = 7 + 1
[2,1,4,3,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? => ?
 => ? = 5 + 1
[5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [7,4,3,5,6,2,1,8] => ?
 => ? = 4 + 1
[1,5,4,7,6,2,3,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,7,4,3,6,5,2,8] => ?
 => ? = 3 + 1
[4,5,3,7,8,6,1,2] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => [8,3,4,2,6,7,5,1] => ?
 => ? = 3 + 1
[4,3,5,6,8,7,1,2] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => [8,3,2,4,5,7,6,1] => ?
 => ? = 3 + 1

Description

The largest part of an integer partition.

Matching statistic: St000010

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 92%

Values

[1] => [1,0]
 => [1] => [1]
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,2] => [2]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [2,1] => [1,1]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => [3]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => [1,1,1]
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => [1,1,1]
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4]
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,1]
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1]
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [3,1]
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,1]
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1]
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,1,1]
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,1]
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [2,1,1]
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,1,1,1]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,1]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,1]
 => 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,1,1]
 => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [4,1]
 => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,2]
 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [4,1]
 => 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [3,1,1]
 => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [3,1,1]
 => 3 = 2 + 1
[6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [7,5,4,3,2,6,8,1] => ?
 => ? = 5 + 1
[6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [8,7,4,5,3,6,2,1] => ?
 => ? = 5 + 1
[5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [6,5,3,4,2,1,7,8] => ?
 => ? = 4 + 1
[4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [6,3,4,5,2,1,7,8] => ?
 => ? = 3 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,8,4,5,7,6,1] => ?
 => ? = 3 + 1
[4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ?
 => ? = 3 + 1
[3,2,6,5,7,4,8,1] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ?
 => ? = 3 + 1
[4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ?
 => ? = 3 + 1
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,8,4,6,7,5,2] => ?
 => ? = 3 + 1
[2,1,6,5,7,4,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ?
 => ? = 3 + 1
[4,3,5,2,6,1,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7] => ?
 => ? = 3 + 1
[2,4,5,6,7,1,8,3] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,7,3,4,5,6,8,1] => ?
 => ? = 2 + 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,8,4,5,7,6,3,1] => ?
 => ? = 4 + 1
[2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,8,4,7,5,6,3,1] => ?
 => ? = 4 + 1
[2,3,5,6,1,7,8,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,3,6,4,5,7,8,1] => ?
 => ? = 2 + 1
[2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ?
 => ? = 3 + 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] => ?
 => ? = 3 + 1
[2,6,1,3,4,5,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,6,5,4,3,1,7,8] => ?
 => ? = 4 + 1
[3,1,5,6,2,8,4,7] => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => [3,2,6,4,5,8,7,1] => ?
 => ? = 2 + 1
[1,3,5,8,2,4,6,7] => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,3,8,4,7,6,5,2] => ?
 => ? = 4 + 1
[1,3,5,6,2,7,4,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,3,6,4,5,7,2,8] => ?
 => ? = 2 + 1
[3,1,6,2,7,4,8,5] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ?
 => ? = 3 + 1
[3,6,1,7,8,2,4,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
 => [8,2,5,4,6,7,3,1] => ?
 => ? = 4 + 1
[1,3,2,6,4,7,5,8] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,3,2,6,5,7,4,8] => ?
 => ? = 2 + 1
[4,1,5,2,6,3,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7] => ?
 => ? = 3 + 1
[4,1,5,6,8,2,3,7] => [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
 => [8,3,2,4,5,7,6,1] => ?
 => ? = 3 + 1
[4,1,5,2,7,3,8,6] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ?
 => ? = 3 + 1
[4,1,5,2,7,8,3,6] => [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
 => [5,3,2,4,8,6,7,1] => ?
 => ? = 3 + 1
[4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
 => [8,3,4,6,5,2,7,1] => ?
 => ? = 4 + 1
[4,5,1,7,8,2,3,6] => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => [8,3,4,2,6,7,5,1] => ?
 => ? = 3 + 1
[4,5,7,8,1,2,3,6] => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [8,3,4,7,5,6,2,1] => ?
 => ? = 4 + 1
[1,4,2,5,3,7,6,8] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,4,3,5,2,7,6,8] => ?
 => ? = 2 + 1
[1,4,2,5,3,8,6,7] => [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [1,4,3,5,2,8,7,6] => ?
 => ? = 2 + 1
[1,4,2,5,3,6,7,8] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [1,4,3,5,2,6,7,8] => ?
 => ? = 2 + 1
[1,4,5,6,7,2,3,8] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,7,3,4,5,6,2,8] => ?
 => ? = 2 + 1
[4,1,6,2,7,3,8,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ?
 => ? = 3 + 1
[4,6,7,1,8,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
 => [8,3,6,4,5,7,2,1] => ?
 => ? = 4 + 1
[1,4,2,6,3,7,5,8] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
 => [1,4,3,6,5,7,2,8] => ?
 => ? = 2 + 1
[4,7,1,2,3,5,6,8] => ?
 => ? => ?
 => ? = 5 + 1
[1,4,2,7,3,5,6,8] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [1,4,3,7,6,5,2,8] => ?
 => ? = 3 + 1
[1,5,2,6,3,7,4,8] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,6,4,3,5,7,2,8] => ?
 => ? = 3 + 1
[1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,2,6,4,5,3,7,8] => ?
 => ? = 2 + 1
[1,5,2,7,3,4,6,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [1,7,4,3,6,5,2,8] => ?
 => ? = 3 + 1
[1,5,2,8,3,4,6,7] => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,8,4,3,7,6,5,2] => ?
 => ? = 4 + 1
[1,5,8,2,3,4,6,7] => ?
 => ? => ?
 => ? = 5 + 1
[1,6,2,7,3,4,5,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,7,5,4,6,3,2,8] => ?
 => ? = 4 + 1
[1,6,2,3,4,7,5,8] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [1,6,5,4,3,7,2,8] => ?
 => ? = 4 + 1
[6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [8,6,4,5,3,7,2,1] => ?
 => ? = 5 + 1
[4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
 => [8,3,6,5,4,2,7,1] => ?
 => ? = 5 + 1
[5,4,7,2,1,8,3,6] => [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
 => [8,4,3,6,5,2,7,1] => ?
 => ? = 4 + 1

Description

The length of the partition.

Matching statistic: St000013
(load all 12 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 92%

Values

[1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[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 = 0 + 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]
 => 2 = 1 + 1
[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]
 => 2 = 1 + 1
[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]
 => 2 = 1 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[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]
 => 2 = 1 + 1
[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]
 => 2 = 1 + 1
[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]
 => 2 = 1 + 1
[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]
 => 2 = 1 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[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]
 => 3 = 2 + 1
[5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => ? = 3 + 1
[3,2,6,5,7,4,8,1] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => ? = 3 + 1
[4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[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]
 => ? = 4 + 1
[1,4,3,6,5,8,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 2 + 1
[2,1,6,5,7,4,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[1,2,6,7,8,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 3 + 1
[3,2,1,6,5,8,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[3,2,4,1,7,6,8,5] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[2,1,4,3,6,8,7,5] => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 2 + 1
[3,2,5,4,1,8,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[4,3,5,2,6,1,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[3,2,5,4,6,1,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[2,1,4,6,5,3,8,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => ? = 2 + 1
[2,1,5,4,6,3,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[2,4,3,1,6,5,8,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[3,2,4,1,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[3,2,7,6,5,4,1,8] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 4 + 1
[5,4,3,2,7,6,1,8] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,5,4,7,6,1,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[2,1,4,5,8,6,7,3] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
 => ? = 3 + 1
[2,3,6,4,5,1,8,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
 => ? = 4 + 1
[2,4,6,1,7,3,8,5] => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => ? = 3 + 1
[2,4,1,3,7,5,8,6] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 2 + 1
[2,1,5,3,6,4,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
 => ? = 4 + 1
[2,5,1,7,3,4,8,6] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,6,7,1,3,4,8,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 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]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,7,1,3,4,5,6,8] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 5 + 1
[2,1,3,8,4,5,6,7] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 4 + 1
[3,1,4,2,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[3,1,4,2,6,8,5,7] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[3,1,4,2,7,5,8,6] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => ? = 2 + 1
[3,1,5,2,6,4,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[3,1,5,6,2,8,4,7] => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[1,3,5,8,2,4,6,7] => [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]
 => ? = 4 + 1
[1,3,5,6,2,7,4,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => ? = 2 + 1

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000451
(load all 3 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 92%

Values

[1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2] => [2,1] => [2,1] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => [2,1] => 2 = 1 + 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,3,1] => [2,3,1] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [1,3,2] => [1,3,2] => [3,1,2] => 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,3,2] => [3,1,2] => 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 3 = 2 + 1
[1,4,3,2] => [4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 3 = 2 + 1
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 1 + 1
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 3 = 2 + 1
[2,4,3,1] => [3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 3 = 2 + 1
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 3 = 2 + 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[3,2,1,4] => [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 3 = 2 + 1
[3,2,4,1] => [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3 = 2 + 1
[3,4,2,1] => [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 3 = 2 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[4,3,1,2] => [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 4 = 3 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => [1,2,5,3,4] => 3 = 2 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,3,2] => [1,2,5,3,4] => 3 = 2 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => [1,3,2,5,4] => 2 = 1 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [5,3,1,4,2] => [1,3,5,2,4] => 3 = 2 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,4,2] => [1,3,5,2,4] => 3 = 2 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => 3 = 2 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,4,3,1] => [1,4,2,3,5] => 3 = 2 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [5,2,4,1,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [1,4,5,2,3] => 3 = 2 + 1
[3,2,4,5,6,7,1] => [5,6,4,3,2,1,7] => [5,7,4,3,2,1,6] => [3,1,4,5,6,7,2] => ? = 2 + 1
[3,4,2,5,6,7,1] => [5,4,6,3,2,1,7] => [5,4,7,3,2,1,6] => [3,4,1,5,6,7,2] => ? = 2 + 1
[3,4,5,2,6,7,1] => [5,4,3,6,2,1,7] => [5,4,3,7,2,1,6] => [3,4,5,1,6,7,2] => ? = 2 + 1
[3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [5,4,3,2,7,1,6] => [3,4,5,6,1,7,2] => ? = 2 + 1
[4,2,3,5,6,7,1] => [4,6,5,3,2,1,7] => [4,7,6,3,2,1,5] => [4,1,2,5,6,7,3] => ? = 3 + 1
[4,2,5,1,3,6,7] => [4,6,3,7,5,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 3 + 1
[4,3,2,5,6,7,1] => [4,5,6,3,2,1,7] => [4,7,6,3,2,1,5] => [4,1,2,5,6,7,3] => ? = 3 + 1
[4,3,5,2,6,7,1] => [4,5,3,6,2,1,7] => [4,7,3,6,2,1,5] => [4,1,5,2,6,7,3] => ? = 3 + 1
[4,5,2,3,6,7,1] => [4,3,6,5,2,1,7] => [4,3,7,6,2,1,5] => [4,5,1,2,6,7,3] => ? = 3 + 1
[5,2,6,7,1,3,4] => [3,6,2,1,7,5,4] => [3,7,2,1,6,5,4] => [5,1,6,7,2,3,4] => ? = 4 + 1
[6,4,7,2,5,1,3] => [2,4,1,6,3,7,5] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 5 + 1
[7,6,8,5,4,3,2,1] => [2,3,1,4,5,6,7,8] => [2,8,1,7,6,5,4,3] => [7,1,8,2,3,4,5,6] => ? = 6 + 1
[7,6,5,4,8,3,2,1] => [2,3,4,5,1,6,7,8] => [2,8,7,6,1,5,4,3] => [7,1,2,3,8,4,5,6] => ? = 6 + 1
[6,5,7,4,8,3,2,1] => [3,4,2,5,1,6,7,8] => [3,8,2,7,1,6,5,4] => [6,1,7,2,8,3,4,5] => ? = 5 + 1
[6,5,7,4,3,2,8,1] => [3,4,2,5,6,7,1,8] => [3,8,2,7,6,5,1,4] => [6,1,7,2,3,4,8,5] => ? = 5 + 1
[6,5,4,3,7,2,8,1] => [3,4,5,6,2,7,1,8] => [3,8,7,6,2,5,1,4] => [6,1,2,3,7,4,8,5] => ? = 5 + 1
[3,4,5,2,6,7,8,1] => [6,5,4,7,3,2,1,8] => [6,5,4,8,3,2,1,7] => [3,4,5,1,6,7,8,2] => ? = 2 + 1
[5,2,3,4,6,7,8,1] => [4,7,6,5,3,2,1,8] => [4,8,7,6,3,2,1,5] => [5,1,2,3,6,7,8,4] => ? = 4 + 1
[4,3,2,5,6,7,8,1] => [5,6,7,4,3,2,1,8] => [5,8,7,4,3,2,1,6] => [4,1,2,5,6,7,8,3] => ? = 3 + 1
[3,4,2,5,6,7,8,1] => [6,5,7,4,3,2,1,8] => [6,5,8,4,3,2,1,7] => [3,4,1,5,6,7,8,2] => ? = 2 + 1
[4,2,3,5,6,7,8,1] => [5,7,6,4,3,2,1,8] => [5,8,7,4,3,2,1,6] => [4,1,2,5,6,7,8,3] => ? = 3 + 1
[3,2,4,5,6,7,8,1] => [6,7,5,4,3,2,1,8] => [6,8,5,4,3,2,1,7] => [3,1,4,5,6,7,8,2] => ? = 2 + 1
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [2,8,1,7,6,5,4,3] => [7,1,8,2,3,4,5,6] => ? = 6 + 1
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [2,8,7,1,6,5,4,3] => [7,1,2,8,3,4,5,6] => ? = 6 + 1
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [2,8,7,1,6,5,4,3] => [7,1,2,8,3,4,5,6] => ? = 6 + 1
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [3,2,8,1,7,6,5,4] => [6,7,1,8,2,3,4,5] => ? = 5 + 1
[6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => [6,1,7,8,2,3,4,5] => ? = 5 + 1
[5,6,3,4,1,2,7,8] => [4,3,6,5,8,7,2,1] => [4,3,8,7,6,5,2,1] => [5,6,1,2,3,4,7,8] => ? = 4 + 1
[4,3,2,1,5,6,7,8] => [5,6,7,8,4,3,2,1] => [5,8,7,6,4,3,2,1] => [4,1,2,3,5,6,7,8] => ? = 3 + 1
[3,2,1,4,5,6,7,8] => [6,7,8,5,4,3,2,1] => [6,8,7,5,4,3,2,1] => [3,1,2,4,5,6,7,8] => ? = 2 + 1
[3,5,8,7,6,4,2,1] => [6,4,1,2,3,5,7,8] => [6,4,1,8,7,5,3,2] => [3,5,8,1,2,4,6,7] => ? = 5 + 1
[4,3,6,5,8,7,2,1] => [5,6,3,4,1,2,7,8] => [5,8,3,7,1,6,4,2] => [4,1,6,2,8,3,5,7] => ? = 3 + 1
[2,6,8,7,5,4,3,1] => [7,3,1,2,4,5,6,8] => [7,3,1,8,6,5,4,2] => [2,6,8,1,3,4,5,7] => ? = 5 + 1
[2,3,5,6,8,7,4,1] => [7,6,4,3,1,2,5,8] => [7,6,4,3,1,8,5,2] => [2,3,5,6,8,1,4,7] => ? = 3 + 1
[3,2,6,5,4,8,7,1] => [6,7,3,4,5,1,2,8] => [6,8,3,7,5,1,4,2] => [3,1,6,2,4,8,5,7] => ? = 3 + 1
[1,3,5,7,8,6,4,2] => [8,6,4,2,1,3,5,7] => [8,6,4,2,1,7,5,3] => [1,3,5,7,8,2,4,6] => ? = 3 + 1
[1,2,6,7,8,5,4,3] => [8,7,3,2,1,4,5,6] => [8,7,3,2,1,6,5,4] => [1,2,6,7,8,3,4,5] => ? = 3 + 1
[3,2,1,8,7,6,5,4] => [6,7,8,1,2,3,4,5] => [6,8,7,1,5,4,3,2] => [3,1,2,8,4,5,6,7] => ? = 4 + 1
[3,2,1,6,5,8,7,4] => [6,7,8,3,4,1,2,5] => [6,8,7,3,5,1,4,2] => [3,1,2,6,4,8,5,7] => ? = 2 + 1
[2,4,3,1,6,8,7,5] => [7,5,6,8,3,1,2,4] => [7,5,8,6,3,1,4,2] => [2,4,1,3,6,8,5,7] => ? = 2 + 1
[2,1,4,3,6,8,7,5] => [7,8,5,6,3,1,2,4] => [7,8,5,6,3,1,4,2] => [2,1,4,3,6,8,5,7] => ? = 2 + 1
[5,4,3,2,1,8,7,6] => [4,5,6,7,8,1,2,3] => [4,8,7,6,5,1,3,2] => [5,1,2,3,4,8,6,7] => ? = 4 + 1
[3,2,5,4,1,8,7,6] => [6,7,4,5,8,1,2,3] => [6,8,4,7,5,1,3,2] => [3,1,5,2,4,8,6,7] => ? = 2 + 1
[2,1,4,6,5,3,8,7] => [7,8,5,3,4,6,1,2] => [7,8,5,3,6,4,1,2] => [2,1,4,6,3,5,8,7] => ? = 2 + 1
[2,4,3,1,6,5,8,7] => [7,5,6,8,3,4,1,2] => [7,5,8,6,3,4,1,2] => [2,4,1,3,6,5,8,7] => ? = 2 + 1
[3,2,7,6,5,4,1,8] => [6,7,2,3,4,5,8,1] => [6,8,2,7,5,4,3,1] => [3,1,7,2,4,5,6,8] => ? = 4 + 1
[5,4,3,2,7,6,1,8] => [4,5,6,7,2,3,8,1] => [4,8,7,6,2,5,3,1] => [5,1,2,3,7,4,6,8] => ? = 4 + 1
[3,2,5,4,7,6,1,8] => [6,7,4,5,2,3,8,1] => [6,8,4,7,2,5,3,1] => [3,1,5,2,7,4,6,8] => ? = 2 + 1
[2,1,4,5,8,6,7,3] => [7,8,5,4,1,3,2,6] => [7,8,5,4,1,6,3,2] => [2,1,4,5,8,3,6,7] => ? = 3 + 1
[2,3,4,8,5,6,7,1] => [7,6,5,1,4,3,2,8] => [7,6,5,1,8,4,3,2] => [2,3,4,8,1,5,6,7] => ? = 4 + 1

Description

The length of the longest pattern of the form k 1 2...(k-1).

Matching statistic: St000097

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%

Values

[1] => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,2] => ([],2)
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,7,3,4,5,6] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[2,1,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[5,1,2,3,4,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[5,4,3,2,1,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [6,3,4,5,2,1,7,8] => ?
 => ? = 3 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,8,4,5,7,6,1] => ?
 => ? = 3 + 1
[3,2,6,5,4,8,7,1] => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [3,2,6,5,4,8,7,1] => ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ?
 => ? = 3 + 1
[3,2,6,5,7,4,8,1] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ?
 => ? = 3 + 1
[4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ?
 => ? = 3 + 1
[2,1,6,5,7,4,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ?
 => ? = 3 + 1
[2,1,5,4,7,6,8,3] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,8,3] => ([(0,7),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,1,8,7,6,5,4] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,2,1,8,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[3,2,1,6,5,8,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [3,2,1,6,5,8,7,4] => ([(0,1),(0,2),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [2,4,3,1,6,8,7,5] => ([(0,7),(1,6),(2,3),(2,6),(3,6),(4,5),(4,7),(5,7)],8)
 => ? = 2 + 1
[2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [2,4,3,1,7,6,8,5] => ([(0,7),(1,6),(2,3),(2,6),(3,6),(4,5),(4,7),(5,7)],8)
 => ? = 2 + 1
[3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
 => [3,2,4,1,6,8,7,5] => ([(0,7),(1,6),(2,3),(2,6),(3,6),(4,5),(4,7),(5,7)],8)
 => ? = 2 + 1
[3,2,4,1,7,6,8,5] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
 => [3,2,4,1,7,6,8,5] => ([(0,7),(1,6),(2,3),(2,6),(3,6),(4,5),(4,7),(5,7)],8)
 => ? = 2 + 1
[2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [2,3,4,1,6,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)
 => ? = 1 + 1
[2,1,4,3,6,8,7,5] => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,3,6,8,7,5] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,1,4,3,7,6,8,5] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,8,5] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[5,4,3,2,1,8,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [5,4,3,2,1,8,7,6] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[3,2,5,4,1,8,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [3,2,5,4,1,8,7,6] => ([(0,1),(0,2),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 2 + 1
[2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,1,4,5,7,8,6] => ([(2,7),(3,7),(4,6),(5,6)],8)
 => ? = 1 + 1
[4,3,5,2,6,1,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [5,3,2,4,6,1,8,7] => ([(0,1),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,2,5,4,6,1,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,5,4,6,1,8,7] => ([(0,7),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,3,4,5,6,1,8,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,5,6,1,8,7] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[2,1,4,6,5,3,8,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,1,4,6,5,3,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,1,5,4,6,3,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,4,3,1,6,5,8,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [2,4,3,1,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,4,1,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => ? = 2 + 1
[1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,6,7,5,8] => ([(2,7),(3,7),(4,6),(5,6)],8)
 => ? = 1 + 1
[2,1,4,5,8,6,7,3] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,4,5,8,7,6,3] => ([(0,1),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[2,3,6,4,5,1,8,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,6,5,4,1,8,7] => ([(0,1),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[4,3,2,8,5,7,6,1] => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => [4,3,2,8,7,6,5,1] => ([(0,1),(0,2),(0,7),(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[2,1,4,3,7,5,8,6] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,7,6,8,5] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,4,1,3,7,5,8,6] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => [2,4,3,1,7,6,8,5] => ([(0,7),(1,6),(2,3),(2,6),(3,6),(4,5),(4,7),(5,7)],8)
 => ? = 2 + 1
[2,4,5,6,7,1,8,3] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [2,7,3,4,5,6,8,1] => ?
 => ? = 2 + 1
[2,1,5,3,6,4,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,1,5,4,6,3,8,7] => ([(0,3),(1,2),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,8,4,5,7,6,3,1] => ?
 => ? = 4 + 1
[2,1,5,3,7,4,8,6] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,5,4,7,6,8,3] => ([(0,7),(1,2),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[2,1,5,7,3,8,4,6] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [2,1,8,4,6,5,7,3] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,8,4,7,5,6,3,1] => ?
 => ? = 4 + 1
[2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,1,7,5,4,6,8,3] => ?
 => ? = 3 + 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] => ?
 => ? = 3 + 1
[2,6,1,3,4,5,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,6,5,4,3,1,7,8] => ?
 => ? = 4 + 1
[2,1,3,8,4,5,6,7] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,8,7,6,5,4] => ([(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St000011

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 92%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[6,5,7,4,8,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5 + 1
[6,5,7,4,3,2,8,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[6,5,4,3,7,2,8,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5 + 1
[5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4 + 1
[4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
[3,5,8,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 5 + 1
[4,3,6,5,8,7,2,1] => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 + 1
[2,6,8,7,5,4,3,1] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[3,2,6,5,4,8,7,1] => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 3 + 1
[3,2,6,5,7,4,8,1] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[1,4,3,6,5,8,7,2] => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[2,1,6,5,7,4,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[1,2,6,7,8,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 3 + 1
[3,2,1,6,5,8,7,4] => [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[2,1,4,3,6,8,7,5] => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,1,4,3,7,6,8,5] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[3,2,5,4,1,8,7,6] => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[4,3,5,2,6,1,8,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[2,1,4,6,5,3,8,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,1,5,4,6,3,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[2,4,3,1,6,5,8,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[3,2,4,1,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[3,2,5,4,7,6,1,8] => [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2 + 1
[2,3,4,8,5,6,7,1] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4 + 1
[2,3,7,4,5,6,8,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4 + 1
[4,3,2,7,5,6,8,1] => [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[2,4,6,1,7,3,8,5] => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[2,1,4,3,7,5,8,6] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,1,4,3,7,8,5,6] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,4,7,8,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 + 1
[2,1,4,3,8,5,6,7] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[2,4,5,6,7,1,8,3] => [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2 + 1
[2,1,5,3,6,4,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[2,5,6,8,1,3,4,7] => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[2,5,1,7,3,4,8,6] => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 3 + 1
[2,5,7,8,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,6,7,1,3,4,8,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4 + 1
[2,1,6,7,8,3,4,5] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[2,6,7,8,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4 + 1
[2,1,6,3,4,5,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[2,6,1,3,4,5,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4 + 1
[2,1,3,8,4,5,6,7] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4 + 1

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St001039
(load all 4 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%

Values

[1] => [1,0]
 => [1,0]
 => ? = 0 + 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[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]
 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[6,7,8,5,2,3,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[7,8,3,2,4,5,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
[3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
[2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 1 + 1
[7,8,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 4 + 1
[7,8,3,4,5,6,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 2 + 1
[6,7,8,5,4,1,2,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[7,8,4,5,6,1,2,3] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[7,8,5,6,1,2,3,4] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 4 + 1
[6,7,8,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[7,8,5,4,2,1,3,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[7,8,5,4,1,2,3,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[7,8,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 6 + 1
[6,7,5,4,3,2,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[5,6,7,4,3,2,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[6,7,5,4,2,3,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[5,6,7,2,3,4,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 1 + 1
[5,6,7,1,2,3,4,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[6,7,1,2,3,4,5,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5 + 1
[2,3,4,5,6,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 1 + 1
[5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 4 + 1
[4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[2,3,4,5,1,6,7,8] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 1 + 1
[3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[3,5,8,7,6,4,2,1] => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[2,6,8,7,5,4,3,1] => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 3 + 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]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
[1,6,8,7,5,4,3,2] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => ? = 5 + 1
[1,6,7,8,5,4,3,2] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 4 + 1
[1,5,8,7,6,4,3,2] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 5 + 1
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 3 + 1
[1,2,6,7,8,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 1
[1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
[3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[2,4,3,1,6,8,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 2 + 1

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000098

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 83%

Values

[1] => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,2] => ([],2)
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,7,3,4,5,6] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[2,1,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 1 + 1
[5,1,2,3,4,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[5,4,3,2,1,7,6] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,5,4,3,2,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,6,4,5,3,2,1,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,7,4,3,2,1,8] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [7,6,4,3,5,2,1,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,6,7,4,3,2,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [7,6,3,4,5,2,1,8] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,6,4,5,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,5,4,7,3,2,1,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [7,5,4,3,6,2,1,8] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,6,3,4,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,6,4,3,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,6,3,4,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,5,4,3,6,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,5,4,2,3,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,6,4,5,3,2,1,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,7,3,2,4,1,8] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [7,6,4,3,5,2,1,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,6,7,2,3,4,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [7,6,3,4,5,2,1,8] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,6,7,1,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[5,6,7,1,2,3,4,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [7,6,3,4,5,2,1,8] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,7,1,2,3,4,5,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,6,4,5,3,2,1,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[2,3,4,5,6,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,5,6,1,7,8] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => [6,5,3,4,2,1,7,8] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [6,3,4,5,2,1,7,8] => ?
 => ? = 3 + 1
[6,1,2,3,4,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[2,3,4,5,1,6,7,8] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,4,5,1,6,7,8] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1 + 1
[5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => [4,2,3,1,5,6,7,8] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(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]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? = 1 + 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] => ([],8)
 => ? = 0 + 1
[2,3,5,6,8,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,3,8,4,5,7,6,1] => ?
 => ? = 3 + 1
[3,2,6,5,4,8,7,1] => [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => [3,2,6,5,4,8,7,1] => ([(0,3),(0,7),(1,2),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[4,3,6,5,7,2,8,1] => [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,3,2,5,4,6,8,1] => ?
 => ? = 3 + 1
[3,2,6,5,7,4,8,1] => [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [3,2,7,5,4,6,8,1] => ?
 => ? = 3 + 1
[4,3,5,2,7,6,8,1] => [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [5,3,2,4,7,6,8,1] => ?
 => ? = 3 + 1
[3,2,5,4,7,6,8,1] => [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,7,6,8,1] => ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 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]
 => [1,8,7,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,6,8,7,5,4,3,2] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,8,7,4,6,5,3,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,7,6,8,5,4,3,2] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,8,7,5,4,6,3,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,6,7,8,5,4,3,2] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,8,7,4,5,6,3,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[1,5,8,7,6,4,3,2] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,8,4,7,6,5,3,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,7,6,5,8,4,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,8,6,5,4,7,3,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,3,8,4,6,7,5,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 + 1
[1,4,3,8,7,6,5,2] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,4,3,8,7,6,5,2] => ([(1,2),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000662
(load all 4 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 83%

Values

[1] => [1,0]
 => [1] => 0
[1,2] => [1,0,1,0]
 => [1,2] => 0
[2,1] => [1,1,0,0]
 => [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 1
[3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 2
[1,5,2,3,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,5,4,3,2,7,6] => ? = 3
[2,1,3,4,7,5,6] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,6,5] => ? = 2
[2,1,3,7,4,5,6] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,6,5,4] => ? = 3
[2,1,6,3,4,5,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,5,4,3,7] => ? = 3
[2,3,4,5,7,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => ? = 2
[2,3,4,6,1,7,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,5,7,1] => ? = 2
[2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,6,7,5,1] => ? = 2
[2,3,4,7,1,5,6] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,7,6,5,1] => ? = 3
[2,3,5,6,7,1,4] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,5,6,7,4,1] => ? = 2
[2,4,1,3,5,6,7] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [2,4,3,1,5,6,7] => ? = 2
[2,5,1,3,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [2,5,4,3,1,6,7] => ? = 3
[2,5,7,1,3,4,6] => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [2,5,7,6,4,3,1] => ? = 4
[2,6,1,7,3,4,5] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [2,6,5,7,4,3,1] => ? = 4
[2,6,7,1,3,4,5] => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,6,7,5,4,3,1] => ? = 4
[2,7,1,3,4,5,6] => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => ? = 5
[3,1,2,4,5,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [3,2,1,4,5,7,6] => ? = 2
[3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [3,4,2,1,5,6,7] => ? = 2
[3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 2
[4,1,2,3,5,7,6] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 3
[4,2,5,1,3,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [4,3,5,2,1,6,7] => ? = 3
[4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [4,5,3,2,6,7,1] => ? = 3
[4,6,1,2,3,7,5] => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [4,6,5,3,2,7,1] => ? = 4
[5,1,2,3,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,4,3,2,6,7,1] => ? = 4
[5,1,2,3,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [5,4,3,2,7,6,1] => ? = 4
[5,1,6,2,3,7,4] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [5,4,6,3,2,7,1] => ? = 4
[5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,4,3,2,6,7,1] => ? = 4
[5,2,6,7,1,3,4] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1] => ? = 4
[5,3,2,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [5,4,3,2,6,7,1] => ? = 4
[5,6,1,2,3,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,4,3,2,7,1] => ? = 4
[5,6,7,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 4
[6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 5
[6,1,2,3,7,4,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 5
[6,1,2,7,3,4,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 5
[6,2,3,4,5,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 5
[6,4,3,2,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 5
[6,4,7,2,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 5
[6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 5
[6,7,4,5,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 5
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [6,7,8,5,4,3,2,1] => ? = 5
[7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[6,7,8,5,2,3,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [6,7,8,5,4,3,2,1] => ? = 5
[7,8,3,2,4,5,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => [5,4,3,2,6,7,8,1] => ? = 4
[7,8,6,5,4,3,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6
[5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => ? = 4
[7,8,3,4,5,6,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [7,8,6,5,4,3,2,1] => ? = 6

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

The following 68 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000028The number of stack-sorts needed to sort a permutation. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000527The width of the poset. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000845The maximal number of elements covered by an element in a poset. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000730The maximal arc length of a set partition. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St000053The number of valleys of the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000651The maximal size of a rise in a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001963The tree-depth of a graph. St001717The largest size of an interval in a poset. St000308The height of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000021The number of descents of a permutation. St000080The rank of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001330The hat guessing number of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. 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. St001880The number of 2-Gorenstein indecomposable injective modules 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. St001589The nesting number of a perfect matching. St000317The cycle descent number of a permutation. St001590The crossing number of a perfect matching. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000983The length of the longest alternating subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.