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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 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]
=> [3,1,1] => 3
[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]
=> [3,1,1] => 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 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]
=> [2,2,1] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 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]
=> [4,1] => 4
[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]
=> [4,1] => 4
[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]
=> [4,1] => 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 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]
=> [4,1] => 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 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]
=> [4,1] => 4
[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]
=> [4,1] => 4
[2,4,6,1,3,5,8,7,9] => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? => ? = 1
[2,4,1,6,8,3,5,7,9] => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? => ? = 1
[6,4,3,2,7,1,5,8,9] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 1
[1,6,7,10,9,4,8,2,5,3] => [1,0,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,1,0]
=> ? => ? = 9
[1,6,7,10,9,2,3,8,5,4] => [1,0,1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,0,1,0]
=> ? => ? = 9
[7,8,4,9,2,5,1,3,6,10] => [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> ?
=> ? => ? = 1
[4,1,5,2,6,3,8,7,9] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? => ? = 1
[1,3,5,2,4,7,6,9,8] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? => ? = 2
[6,4,2,1,7,5,3,8,9] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 1
[2,3,5,6,7,1,8,4,9] => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? => ? = 1
[1,8,11,10,9,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? => ? = 10
[1,6,11,10,9,8,7,5,4,3,2] => [1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0]
=> ? => ? = 10
[1,5,11,10,9,8,7,6,4,3,2] => [1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
=> ? => ? = 10
[1,3,11,10,9,8,7,6,5,4,2] => [1,0,1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0]
=> ? => ? = 10
[1,3,5,9,8,7,6,4,2] => [1,0,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,1,0]
=> ? => ? = 8
[1,3,5,7,10,9,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? => ? = 9
[1,3,5,10,9,8,7,6,4,2] => [1,0,1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ?
=> ? => ? = 9
[3,5,1,6,2,4,8,7,9] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? => ? = 1
[3,5,1,2,6,4,8,7,9] => [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? => ? = 1
[4,2,1,6,3,5,8,7,9] => [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? => ? = 1
[6,4,2,7,1,3,5,8,9] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? => ? = 1
[2,4,3,1,6,8,7,5,10,9,12,11] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ?
=> ? => ? = 2
[1,5,6,9,2,3,8,7,4] => [1,0,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ? = 8
[1,5,6,10,2,3,9,8,7,4] => [1,0,1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0,0,1,0]
=> ? => ? = 9
[6,4,2,8,1,3,5,7,9] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? => ? = 1
[4,2,5,1,6,3,8,7,9] => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? => ? = 1
[1,3,5,2,7,4,6,9,8] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ? = 2
[1,3,5,2,7,4,9,6,8] => [1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? => ? = 8
[3,5,2,7,1,8,6,4,9] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? => ? = 1
[1,6,5,7,4,8,3,9,2] => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? => ? = 8
[1,7,9,10,2,5,8,3,6,4] => [1,0,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> ?
=> ? => ? = 9
[4,1,2,6,5,3,8,7,9] => [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? => ? = 1
[4,1,6,2,3,5,8,7,9] => [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? => ? = 1
[1,3,5,2,6,9,7,10,4,8] => [1,0,1,1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? => ? = 9
[5,4,3,2,1,7,8,9,10,6] => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ?
=> ? => ? = 5
[1,6,2,7,3,8,4,9,5] => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? => ? = 8
[3,4,5,6,1,7,2,8,9] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? => ? = 1
[4,5,3,6,2,7,1,8,9] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? => ? = 1
[1,3,4,5,2,7,6,9,8] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? => ? = 2
[6,5,4,3,7,1,2,8,9] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? => ? = 1
[1,5,6,7,4,8,3,9,2] => [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? => ? = 8
[2,3,4,5,8,7,6,1,9] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[1,3,4,5,6,9,8,7,2] => [1,0,1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ? = 8
[1,3,4,5,6,7,8,9,11,2,10] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? => ? = 10
[2,3,4,8,5,6,7,1,9] => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 1
[3,1,4,6,2,5,8,7,9] => [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? => ? = 1
[4,2,6,1,3,5,8,7,9] => [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? => ? = 1
[1,5,2,4,7,6,9,8,3] => [1,0,1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? => ? = 8
[1,6,2,7,4,8,5,9,3] => [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? => ? = 8
Description
The first part of an integer composition.
Matching statistic: St000645
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[5,4,6,7,3,2,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 1 - 1
[5,4,6,7,2,3,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 1 - 1
[5,6,4,3,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [8,6,2,1,3,4,5,7] => ?
=> ? = 1 - 1
[5,4,6,3,2,7,1,8] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,1,2,4,5,7] => ?
=> ? = 1 - 1
[5,6,3,4,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [8,6,2,1,3,4,5,7] => ?
=> ? = 1 - 1
[5,4,6,2,3,7,1,8] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [8,6,3,1,2,4,5,7] => ?
=> ? = 1 - 1
[5,4,6,7,3,1,2,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 1 - 1
[4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,1,2,6,7] => ?
=> ? = 1 - 1
[5,4,6,7,2,1,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 1 - 1
[5,4,6,7,1,2,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 1 - 1
[5,4,6,3,2,1,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 1 - 1
[5,4,6,2,3,1,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 1 - 1
[3,4,2,5,6,1,7,8] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,2,1,3,6] => ?
=> ? = 1 - 1
[5,4,6,3,1,2,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 1 - 1
[4,3,5,6,1,2,7,8] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,1,2,5,6] => ?
=> ? = 1 - 1
[5,4,6,2,1,3,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 1 - 1
[1,4,8,7,6,5,3,2] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[1,4,7,8,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,2,7,8,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[1,4,6,8,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ?
=> ? = 7 - 1
[1,4,6,7,8,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,2,7,8,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[1,5,4,7,8,6,3,2] => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,3,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 7 - 1
[1,4,5,7,8,6,3,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,2,7,8,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 7 - 1
[1,6,5,4,8,7,3,2] => [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,6,2,3,4,7,8,1] => ?
=> ? = 7 - 1
[1,4,6,5,8,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,2,7,8,1] => ?
=> ? = 7 - 1
[1,5,4,6,8,7,3,2] => [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,2,3,7,8,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 7 - 1
[1,4,6,7,5,8,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,7,8,1] => ?
=> ? = 7 - 1
[1,3,7,8,6,5,4,2] => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,2,8,1] => ?
=> ? = 7 - 1
[1,3,6,8,7,5,4,2] => [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[1,3,6,7,8,5,4,2] => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,2,8,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[1,3,5,8,7,6,4,2] => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,2,8,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 7 - 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]
=> [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 7 - 1
[1,3,5,6,8,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 7 - 1
[1,4,3,7,8,6,5,2] => [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,2,3,8,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 7 - 1
[1,3,4,6,7,8,5,2] => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,3,2,8,1] => ?
=> ? = 7 - 1
[1,4,5,3,8,7,6,2] => [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => ?
=> ? = 7 - 1
[1,3,5,4,8,7,6,2] => [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ? = 7 - 1
[1,3,4,5,8,7,6,2] => [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,3,2,8,1] => ?
=> ? = 7 - 1
[1,4,6,5,3,8,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,2,5,8,1] => ?
=> ? = 7 - 1
[1,3,6,5,4,8,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ? = 7 - 1
[1,5,4,3,6,8,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [6,7,5,2,3,4,8,1] => ?
=> ? = 7 - 1
[1,4,6,5,7,3,8,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ?
=> ? = 7 - 1
[1,3,4,6,7,5,8,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ?
=> ? = 7 - 1
[1,4,5,3,2,7,8,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,2,4,5,1] => ?
=> ? = 3 - 1
[2,1,3,5,4,7,8,6] => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,1,2] => ?
=> ? = 3 - 1
[1,3,6,5,4,2,8,7] => [1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,2,6,1] => ?
=> ? = 2 - 1
[2,3,1,6,5,4,8,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,2,1,3] => ?
=> ? = 2 - 1
[1,3,2,5,6,4,8,7] => [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => ?
=> ? = 2 - 1
[2,1,3,5,6,4,8,7] => [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,1,2] => ?
=> ? = 2 - 1
[1,2,3,6,5,4,8,7] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 2 - 1
[1,2,3,5,6,4,8,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ? = 2 - 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? = 1
[1,2] => 1 => 1 => 1
[2,1] => 0 => 0 => 2
[1,2,3] => 11 => 11 => 1
[1,3,2] => 10 => 01 => 2
[2,1,3] => 01 => 10 => 1
[2,3,1] => 00 => 00 => 3
[3,1,2] => 00 => 00 => 3
[3,2,1] => 00 => 00 => 3
[1,2,3,4] => 111 => 111 => 1
[1,2,4,3] => 110 => 011 => 2
[1,3,2,4] => 101 => 101 => 1
[1,3,4,2] => 100 => 001 => 3
[1,4,2,3] => 100 => 001 => 3
[1,4,3,2] => 100 => 001 => 3
[2,1,3,4] => 011 => 110 => 1
[2,1,4,3] => 010 => 010 => 2
[2,3,1,4] => 001 => 100 => 1
[2,3,4,1] => 000 => 000 => 4
[2,4,1,3] => 000 => 000 => 4
[2,4,3,1] => 000 => 000 => 4
[3,1,2,4] => 001 => 100 => 1
[3,1,4,2] => 000 => 000 => 4
[3,2,1,4] => 001 => 100 => 1
[3,2,4,1] => 000 => 000 => 4
[3,4,1,2] => 000 => 000 => 4
[3,4,2,1] => 000 => 000 => 4
[4,1,2,3] => 000 => 000 => 4
[4,1,3,2] => 000 => 000 => 4
[4,2,1,3] => 000 => 000 => 4
[4,2,3,1] => 000 => 000 => 4
[4,3,1,2] => 000 => 000 => 4
[4,3,2,1] => 000 => 000 => 4
[1,2,3,4,5] => 1111 => 1111 => 1
[1,2,3,5,4] => 1110 => 0111 => 2
[1,2,4,3,5] => 1101 => 1011 => 1
[1,2,4,5,3] => 1100 => 0011 => 3
[1,2,5,3,4] => 1100 => 0011 => 3
[1,2,5,4,3] => 1100 => 0011 => 3
[1,3,2,4,5] => 1011 => 1101 => 1
[1,3,2,5,4] => 1010 => 0101 => 2
[1,3,4,2,5] => 1001 => 1001 => 1
[1,3,4,5,2] => 1000 => 0001 => 4
[1,3,5,2,4] => 1000 => 0001 => 4
[1,3,5,4,2] => 1000 => 0001 => 4
[1,4,2,3,5] => 1001 => 1001 => 1
[1,4,2,5,3] => 1000 => 0001 => 4
[1,4,3,2,5] => 1001 => 1001 => 1
[1,4,3,5,2] => 1000 => 0001 => 4
[1,4,5,2,3] => 1000 => 0001 => 4
[1,4,5,3,2] => 1000 => 0001 => 4
[6,5,7,3,4,2,1,8] => ? => ? => ? = 1
[7,4,5,3,6,2,1,8] => ? => ? => ? = 1
[7,4,5,6,2,3,1,8] => ? => ? => ? = 1
[7,4,2,3,5,6,1,8] => ? => ? => ? = 1
[5,3,4,6,2,7,1,8] => ? => ? => ? = 1
[4,3,5,6,2,7,1,8] => ? => ? => ? = 1
[5,4,6,2,3,7,1,8] => ? => ? => ? = 1
[6,5,3,2,4,7,1,8] => ? => ? => ? = 1
[6,4,2,3,5,7,1,8] => ? => ? => ? = 1
[4,5,2,3,6,7,1,8] => ? => ? => ? = 1
[7,4,5,6,3,1,2,8] => ? => ? => ? = 1
[7,5,6,3,4,1,2,8] => ? => ? => ? = 1
[7,5,3,4,6,1,2,8] => ? => ? => ? = 1
[6,4,3,5,7,1,2,8] => ? => ? => ? = 1
[6,3,4,5,7,1,2,8] => ? => ? => ? = 1
[7,5,4,6,2,1,3,8] => ? => ? => ? = 1
[7,4,5,6,2,1,3,8] => ? => ? => ? = 1
[5,4,6,7,2,1,3,8] => ? => ? => ? = 1
[7,5,6,4,1,2,3,8] => ? => ? => ? = 1
[7,6,4,5,1,2,3,8] => ? => ? => ? = 1
[7,5,4,6,1,2,3,8] => ? => ? => ? = 1
[4,5,6,7,1,2,3,8] => ? => ? => ? = 1
[6,7,5,2,3,1,4,8] => ? => ? => ? = 1
[7,5,6,2,3,1,4,8] => ? => ? => ? = 1
[6,5,7,2,3,1,4,8] => ? => ? => ? = 1
[5,6,7,2,3,1,4,8] => ? => ? => ? = 1
[7,5,6,3,1,2,4,8] => ? => ? => ? = 1
[7,5,6,2,1,3,4,8] => ? => ? => ? = 1
[6,7,4,2,3,1,5,8] => ? => ? => ? = 1
[7,6,2,3,1,4,5,8] => ? => ? => ? = 1
[7,4,2,3,5,1,6,8] => ? => ? => ? = 1
[7,5,3,4,1,2,6,8] => ? => ? => ? = 1
[7,3,4,1,2,5,6,8] => ? => ? => ? = 1
[6,3,4,5,2,1,7,8] => ? => ? => ? = 1
[5,4,6,2,3,1,7,8] => ? => ? => ? = 1
[5,6,3,2,4,1,7,8] => ? => ? => ? = 1
[4,3,5,2,6,1,7,8] => ? => ? => ? = 1
[6,4,5,2,1,3,7,8] => ? => ? => ? = 1
[6,5,1,2,3,4,7,8] => ? => ? => ? = 1
[6,3,2,1,4,5,7,8] => ? => ? => ? = 1
[6,2,1,3,4,5,7,8] => ? => ? => ? = 1
[1,4,5,7,8,6,3,2] => ? => ? => ? = 7
[1,6,5,4,8,7,3,2] => ? => ? => ? = 7
[1,5,4,6,8,7,3,2] => ? => ? => ? = 7
[1,7,6,5,4,8,3,2] => ? => ? => ? = 7
[1,4,6,7,5,8,3,2] => ? => ? => ? = 7
[1,3,6,7,5,8,4,2] => ? => ? => ? = 7
[1,3,5,7,6,8,4,2] => ? => ? => ? = 7
[1,5,4,3,8,7,6,2] => ? => ? => ? = 7
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1] => => => => ? = 1 - 1
[1,2] => 1 => 1 => 0 => 0 = 1 - 1
[2,1] => 0 => 0 => 1 => 1 = 2 - 1
[1,2,3] => 11 => 11 => 00 => 0 = 1 - 1
[1,3,2] => 10 => 01 => 10 => 1 = 2 - 1
[2,1,3] => 01 => 10 => 01 => 0 = 1 - 1
[2,3,1] => 00 => 00 => 11 => 2 = 3 - 1
[3,1,2] => 00 => 00 => 11 => 2 = 3 - 1
[3,2,1] => 00 => 00 => 11 => 2 = 3 - 1
[1,2,3,4] => 111 => 111 => 000 => 0 = 1 - 1
[1,2,4,3] => 110 => 011 => 100 => 1 = 2 - 1
[1,3,2,4] => 101 => 101 => 010 => 0 = 1 - 1
[1,3,4,2] => 100 => 001 => 110 => 2 = 3 - 1
[1,4,2,3] => 100 => 001 => 110 => 2 = 3 - 1
[1,4,3,2] => 100 => 001 => 110 => 2 = 3 - 1
[2,1,3,4] => 011 => 110 => 001 => 0 = 1 - 1
[2,1,4,3] => 010 => 010 => 101 => 1 = 2 - 1
[2,3,1,4] => 001 => 100 => 011 => 0 = 1 - 1
[2,3,4,1] => 000 => 000 => 111 => 3 = 4 - 1
[2,4,1,3] => 000 => 000 => 111 => 3 = 4 - 1
[2,4,3,1] => 000 => 000 => 111 => 3 = 4 - 1
[3,1,2,4] => 001 => 100 => 011 => 0 = 1 - 1
[3,1,4,2] => 000 => 000 => 111 => 3 = 4 - 1
[3,2,1,4] => 001 => 100 => 011 => 0 = 1 - 1
[3,2,4,1] => 000 => 000 => 111 => 3 = 4 - 1
[3,4,1,2] => 000 => 000 => 111 => 3 = 4 - 1
[3,4,2,1] => 000 => 000 => 111 => 3 = 4 - 1
[4,1,2,3] => 000 => 000 => 111 => 3 = 4 - 1
[4,1,3,2] => 000 => 000 => 111 => 3 = 4 - 1
[4,2,1,3] => 000 => 000 => 111 => 3 = 4 - 1
[4,2,3,1] => 000 => 000 => 111 => 3 = 4 - 1
[4,3,1,2] => 000 => 000 => 111 => 3 = 4 - 1
[4,3,2,1] => 000 => 000 => 111 => 3 = 4 - 1
[1,2,3,4,5] => 1111 => 1111 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 1110 => 0111 => 1000 => 1 = 2 - 1
[1,2,4,3,5] => 1101 => 1011 => 0100 => 0 = 1 - 1
[1,2,4,5,3] => 1100 => 0011 => 1100 => 2 = 3 - 1
[1,2,5,3,4] => 1100 => 0011 => 1100 => 2 = 3 - 1
[1,2,5,4,3] => 1100 => 0011 => 1100 => 2 = 3 - 1
[1,3,2,4,5] => 1011 => 1101 => 0010 => 0 = 1 - 1
[1,3,2,5,4] => 1010 => 0101 => 1010 => 1 = 2 - 1
[1,3,4,2,5] => 1001 => 1001 => 0110 => 0 = 1 - 1
[1,3,4,5,2] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,3,5,2,4] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,3,5,4,2] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,4,2,3,5] => 1001 => 1001 => 0110 => 0 = 1 - 1
[1,4,2,5,3] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,4,3,2,5] => 1001 => 1001 => 0110 => 0 = 1 - 1
[1,4,3,5,2] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,4,5,2,3] => 1000 => 0001 => 1110 => 3 = 4 - 1
[1,4,5,3,2] => 1000 => 0001 => 1110 => 3 = 4 - 1
[6,5,7,3,4,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,4,5,3,6,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,4,5,6,2,3,1,8] => ? => ? => ? => ? = 1 - 1
[7,4,2,3,5,6,1,8] => ? => ? => ? => ? = 1 - 1
[5,3,4,6,2,7,1,8] => ? => ? => ? => ? = 1 - 1
[4,3,5,6,2,7,1,8] => ? => ? => ? => ? = 1 - 1
[5,4,6,2,3,7,1,8] => ? => ? => ? => ? = 1 - 1
[6,5,3,2,4,7,1,8] => ? => ? => ? => ? = 1 - 1
[6,4,2,3,5,7,1,8] => ? => ? => ? => ? = 1 - 1
[4,5,2,3,6,7,1,8] => ? => ? => ? => ? = 1 - 1
[7,4,5,6,3,1,2,8] => ? => ? => ? => ? = 1 - 1
[7,5,6,3,4,1,2,8] => ? => ? => ? => ? = 1 - 1
[7,5,3,4,6,1,2,8] => ? => ? => ? => ? = 1 - 1
[6,4,3,5,7,1,2,8] => ? => ? => ? => ? = 1 - 1
[6,3,4,5,7,1,2,8] => ? => ? => ? => ? = 1 - 1
[7,5,4,6,2,1,3,8] => ? => ? => ? => ? = 1 - 1
[7,4,5,6,2,1,3,8] => ? => ? => ? => ? = 1 - 1
[5,4,6,7,2,1,3,8] => ? => ? => ? => ? = 1 - 1
[7,5,6,4,1,2,3,8] => ? => ? => ? => ? = 1 - 1
[7,6,4,5,1,2,3,8] => ? => ? => ? => ? = 1 - 1
[7,5,4,6,1,2,3,8] => ? => ? => ? => ? = 1 - 1
[4,5,6,7,1,2,3,8] => ? => ? => ? => ? = 1 - 1
[6,7,5,2,3,1,4,8] => ? => ? => ? => ? = 1 - 1
[7,5,6,2,3,1,4,8] => ? => ? => ? => ? = 1 - 1
[6,5,7,2,3,1,4,8] => ? => ? => ? => ? = 1 - 1
[5,6,7,2,3,1,4,8] => ? => ? => ? => ? = 1 - 1
[7,5,6,3,1,2,4,8] => ? => ? => ? => ? = 1 - 1
[7,5,6,2,1,3,4,8] => ? => ? => ? => ? = 1 - 1
[6,7,4,2,3,1,5,8] => ? => ? => ? => ? = 1 - 1
[7,6,2,3,1,4,5,8] => ? => ? => ? => ? = 1 - 1
[7,4,2,3,5,1,6,8] => ? => ? => ? => ? = 1 - 1
[7,5,3,4,1,2,6,8] => ? => ? => ? => ? = 1 - 1
[7,3,4,1,2,5,6,8] => ? => ? => ? => ? = 1 - 1
[6,3,4,5,2,1,7,8] => ? => ? => ? => ? = 1 - 1
[5,4,6,2,3,1,7,8] => ? => ? => ? => ? = 1 - 1
[5,6,3,2,4,1,7,8] => ? => ? => ? => ? = 1 - 1
[4,3,5,2,6,1,7,8] => ? => ? => ? => ? = 1 - 1
[6,4,5,2,1,3,7,8] => ? => ? => ? => ? = 1 - 1
[6,5,1,2,3,4,7,8] => ? => ? => ? => ? = 1 - 1
[6,3,2,1,4,5,7,8] => ? => ? => ? => ? = 1 - 1
[6,2,1,3,4,5,7,8] => ? => ? => ? => ? = 1 - 1
[1,4,5,7,8,6,3,2] => ? => ? => ? => ? = 7 - 1
[1,6,5,4,8,7,3,2] => ? => ? => ? => ? = 7 - 1
[1,5,4,6,8,7,3,2] => ? => ? => ? => ? = 7 - 1
[1,7,6,5,4,8,3,2] => ? => ? => ? => ? = 7 - 1
[1,4,6,7,5,8,3,2] => ? => ? => ? => ? = 7 - 1
[1,3,6,7,5,8,4,2] => ? => ? => ? => ? = 7 - 1
[1,3,5,7,6,8,4,2] => ? => ? => ? => ? = 7 - 1
[1,5,4,3,8,7,6,2] => ? => ? => ? => ? = 7 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000007
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[3,1,2] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4
[2,4,1,3] => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 4
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,1,4,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[4,1,2,3] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4
[4,1,3,2] => [4,2,3,1] => [4,3,2,1] => [4,3,2,1] => 4
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 3
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 3
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 3
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,5,7,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,6,4,5,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 1
[1,2,3,6,5,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,6,7,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,7,4,5,6] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,7,4,6,5] => [1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,7,5,4,6] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,7,5,6,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,7,6,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 4
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 1
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 3
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 1
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 2
[1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 1
[1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,5,7,3,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,5,7,6,3] => [1,2,7,4,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 1
[1,2,4,6,3,7,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,6,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 1
[1,2,4,6,5,7,3] => [1,2,7,5,4,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,6,7,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,6,7,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,7,3,5,6] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,7,3,6,5] => [1,2,5,7,3,6,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,7,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,7,6,3,5] => [1,2,6,7,5,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,4,7,6,5,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 2
[1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 1
[1,2,5,3,6,7,4] => [1,2,7,4,5,6,3] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
[1,2,5,3,7,4,6] => [1,2,6,4,7,3,5] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 5
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000505
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> {{1}}
=> 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 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]
=> {{1,3},{2},{4},{5}}
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 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]
=> {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 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]
=> {{1,4},{2},{3},{5}}
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 4
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 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]
=> {{1,3,4},{2},{5}}
=> 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 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]
=> {{1,3,4},{2},{5}}
=> 4
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 4
[5,6,4,7,3,2,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,6,8},{5},{7}}
=> ? = 1
[5,4,6,7,3,2,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 1
[4,5,6,7,3,2,1,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5},{6},{7}}
=> ? = 1
[5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,5,6,8},{4},{7}}
=> ? = 1
[5,4,6,3,7,2,1,8] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,5,7,8},{4},{6}}
=> ? = 1
[5,4,3,6,7,2,1,8] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> {{1},{2,3,6,7,8},{4},{5}}
=> ? = 1
[3,4,5,6,7,2,1,8] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1},{2,3,8},{4},{5},{6},{7}}
=> ? = 1
[5,6,4,7,2,3,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,6,8},{5},{7}}
=> ? = 1
[5,4,6,7,2,3,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 1
[4,5,6,7,2,3,1,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5},{6},{7}}
=> ? = 1
[5,6,4,3,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,4,5,6,8},{3},{7}}
=> ? = 1
[5,4,6,3,2,7,1,8] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,4,5,7,8},{3},{6}}
=> ? = 1
[5,6,3,4,2,7,1,8] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,4,5,6,8},{3},{7}}
=> ? = 1
[5,4,3,6,2,7,1,8] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> {{1},{2,4,6,7,8},{3},{5}}
=> ? = 1
[4,5,3,6,2,7,1,8] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 1
[5,3,4,6,2,7,1,8] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> {{1},{2,4,6,7,8},{3},{5}}
=> ? = 1
[4,3,5,6,2,7,1,8] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> {{1},{2,4,7,8},{3},{5},{6}}
=> ? = 1
[3,4,5,6,2,7,1,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> {{1},{2,4,8},{3},{5},{6},{7}}
=> ? = 1
[5,4,6,2,3,7,1,8] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2,4,5,7,8},{3},{6}}
=> ? = 1
[5,4,3,2,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3},{4}}
=> ? = 1
[5,4,2,3,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3},{4}}
=> ? = 1
[4,5,2,3,6,7,1,8] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1},{2,5,6,8},{3},{4},{7}}
=> ? = 1
[5,2,3,4,6,7,1,8] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2,5,6,7,8},{3},{4}}
=> ? = 1
[4,3,2,5,6,7,1,8] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1},{2,6,7,8},{3},{4},{5}}
=> ? = 1
[3,4,2,5,6,7,1,8] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> {{1},{2,6,8},{3},{4},{5},{7}}
=> ? = 1
[4,2,3,5,6,7,1,8] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1},{2,6,7,8},{3},{4},{5}}
=> ? = 1
[3,2,4,5,6,7,1,8] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1},{2,7,8},{3},{4},{5},{6}}
=> ? = 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]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> ? = 1
[5,4,6,7,3,1,2,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 1
[4,5,6,7,3,1,2,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5},{6},{7}}
=> ? = 1
[5,6,3,4,7,1,2,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,5,6,8},{4},{7}}
=> ? = 1
[4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> {{1},{2,3,7,8},{4},{5},{6}}
=> ? = 1
[3,4,5,6,7,1,2,8] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1},{2,3,8},{4},{5},{6},{7}}
=> ? = 1
[5,4,6,7,2,1,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 1
[4,5,6,7,2,1,3,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5},{6},{7}}
=> ? = 1
[5,6,4,7,1,2,3,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,6,8},{5},{7}}
=> ? = 1
[5,4,6,7,1,2,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> {{1},{2,3,4,7,8},{5},{6}}
=> ? = 1
[4,5,6,7,1,2,3,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,8},{5},{6},{7}}
=> ? = 1
[5,6,4,3,2,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 1
[5,4,6,3,2,1,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,7,8},{6}}
=> ? = 1
[4,5,6,3,2,1,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> {{1},{2},{3,4,5,8},{6},{7}}
=> ? = 1
[5,6,3,4,2,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 1
[5,4,3,6,2,1,7,8] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> {{1},{2},{3,4,6,7,8},{5}}
=> ? = 1
[3,4,5,6,2,1,7,8] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> {{1},{2},{3,4,8},{5},{6},{7}}
=> ? = 1
[5,4,6,2,3,1,7,8] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,7,8},{6}}
=> ? = 1
[5,6,3,2,4,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 1
[5,6,2,3,4,1,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,8},{7}}
=> ? = 1
[5,4,3,2,6,1,7,8] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3,5,6,7,8},{4}}
=> ? = 1
[4,5,3,2,6,1,7,8] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> {{1},{2},{3,5,6,8},{4},{7}}
=> ? = 1
[4,3,5,2,6,1,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> {{1},{2},{3,5,7,8},{4},{6}}
=> ? = 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000026
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 70%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 70%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 2
[1,2,3] => [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
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [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
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[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
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[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]
=> 3
[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
[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
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[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
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[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]
=> 4
[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]
=> 4
[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]
=> 4
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[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]
=> 4
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[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]
=> 4
[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]
=> 4
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[7,5,6,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,7,4,5,3,2,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[7,5,4,6,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[7,4,5,6,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,6,4,7,3,2,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 1
[5,4,6,7,3,2,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[4,5,6,7,3,2,1,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[7,6,5,3,4,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,7,5,3,4,2,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,5,7,3,4,2,1,8] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[5,6,7,3,4,2,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,7,4,3,5,2,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[7,4,5,3,6,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[7,3,4,5,6,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,5,4,3,7,2,1,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[5,4,6,3,7,2,1,8] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[5,4,3,6,7,2,1,8] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[3,4,5,6,7,2,1,8] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[7,6,5,4,2,3,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[7,5,6,4,2,3,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,6,7,4,2,3,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[7,6,4,5,2,3,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,7,4,5,2,3,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[7,5,4,6,2,3,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[7,4,5,6,2,3,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,5,4,7,2,3,1,8] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,6,4,7,2,3,1,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 1
[5,4,6,7,2,3,1,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[4,5,6,7,2,3,1,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[7,6,5,3,2,4,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[5,6,7,3,2,4,1,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[7,6,5,2,3,4,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,7,5,2,3,4,1,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[7,5,6,2,3,4,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[6,5,7,2,3,4,1,8] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 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]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1
[7,6,4,3,2,5,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000273
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 70%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 70%
Values
[1] => => [1] => ([],1)
=> 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1] => 0 => [2] => ([],2)
=> 2
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => 00 => [3] => ([],3)
=> 3
[3,1,2] => 00 => [3] => ([],3)
=> 3
[3,2,1] => 00 => [3] => ([],3)
=> 3
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
=> 3
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 3
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 3
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,3,4,1] => 000 => [4] => ([],4)
=> 4
[2,4,1,3] => 000 => [4] => ([],4)
=> 4
[2,4,3,1] => 000 => [4] => ([],4)
=> 4
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => 000 => [4] => ([],4)
=> 4
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => 000 => [4] => ([],4)
=> 4
[3,4,1,2] => 000 => [4] => ([],4)
=> 4
[3,4,2,1] => 000 => [4] => ([],4)
=> 4
[4,1,2,3] => 000 => [4] => ([],4)
=> 4
[4,1,3,2] => 000 => [4] => ([],4)
=> 4
[4,2,1,3] => 000 => [4] => ([],4)
=> 4
[4,2,3,1] => 000 => [4] => ([],4)
=> 4
[4,3,1,2] => 000 => [4] => ([],4)
=> 4
[4,3,2,1] => 000 => [4] => ([],4)
=> 4
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,2,4,5,3,6,7] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,5,3,4,6,7] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,5,4,3,6,7] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[7,6,5,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,7,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,7,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,5,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,4,5,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,4,6,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,4,5,6,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,4,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,4,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,7,3,4,2,1,8] => ? => ? => ?
=> ? = 1
[5,6,7,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,3,5,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,4,3,5,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,3,4,5,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,4,3,6,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,4,5,3,6,2,1,8] => ? => ? => ?
=> ? = 1
[7,3,4,5,6,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,4,3,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,4,3,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,3,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,3,6,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[3,4,5,6,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,7,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,5,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,4,5,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,4,6,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,4,5,6,2,3,1,8] => ? => ? => ?
=> ? = 1
[6,5,4,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,4,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,3,2,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,7,3,2,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,7,3,2,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,2,3,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,2,3,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,2,3,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
Matching statistic: St000544
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 70%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000544: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 70%
Values
[1] => => [1] => ([],1)
=> 1
[1,2] => 1 => [1,1] => ([(0,1)],2)
=> 1
[2,1] => 0 => [2] => ([],2)
=> 2
[1,2,3] => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => 10 => [1,2] => ([(1,2)],3)
=> 2
[2,1,3] => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => 00 => [3] => ([],3)
=> 3
[3,1,2] => 00 => [3] => ([],3)
=> 3
[3,2,1] => 00 => [3] => ([],3)
=> 3
[1,2,3,4] => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => 100 => [1,3] => ([(2,3)],4)
=> 3
[1,4,2,3] => 100 => [1,3] => ([(2,3)],4)
=> 3
[1,4,3,2] => 100 => [1,3] => ([(2,3)],4)
=> 3
[2,1,3,4] => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3] => 010 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,3,4,1] => 000 => [4] => ([],4)
=> 4
[2,4,1,3] => 000 => [4] => ([],4)
=> 4
[2,4,3,1] => 000 => [4] => ([],4)
=> 4
[3,1,2,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,4,2] => 000 => [4] => ([],4)
=> 4
[3,2,1,4] => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => 000 => [4] => ([],4)
=> 4
[3,4,1,2] => 000 => [4] => ([],4)
=> 4
[3,4,2,1] => 000 => [4] => ([],4)
=> 4
[4,1,2,3] => 000 => [4] => ([],4)
=> 4
[4,1,3,2] => 000 => [4] => ([],4)
=> 4
[4,2,1,3] => 000 => [4] => ([],4)
=> 4
[4,2,3,1] => 000 => [4] => ([],4)
=> 4
[4,3,1,2] => 000 => [4] => ([],4)
=> 4
[4,3,2,1] => 000 => [4] => ([],4)
=> 4
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,4,3,5] => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,2,5,4,3] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,4,5] => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,3,5,2,4] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,3,5,4,2] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,4,2,3,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,4,3,2,5] => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,4,5,2,3] => 1000 => [1,4] => ([(3,4)],5)
=> 4
[1,2,4,5,3,6,7] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,5,3,4,6,7] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,5,4,3,6,7] => 110011 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[7,6,5,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,7,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,7,4,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,5,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,4,5,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,4,6,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,4,5,6,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,4,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,4,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,3,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,7,3,4,2,1,8] => ? => ? => ?
=> ? = 1
[5,6,7,3,4,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,3,5,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,4,3,5,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,3,4,5,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,4,3,6,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,4,5,3,6,2,1,8] => ? => ? => ?
=> ? = 1
[7,3,4,5,6,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,4,3,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,4,3,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,3,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,3,6,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[3,4,5,6,7,2,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,7,4,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,4,5,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,4,5,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,4,6,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,4,5,6,2,3,1,8] => ? => ? => ?
=> ? = 1
[6,5,4,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,4,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,4,6,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[4,5,6,7,2,3,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,3,2,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,5,7,3,2,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[5,6,7,3,2,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,6,5,2,3,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[6,7,5,2,3,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[7,5,6,2,3,4,1,8] => 0000001 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
Description
The cop number of a graph.
This is the minimal number of cops needed to catch the robber. The algorithm is from [2].
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001645The pebbling number of a connected graph. St000740The last entry of a permutation. St001330The hat guessing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000287The number of connected components of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000199The column of the unique '1' in the last row of the alternating sign matrix.
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