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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St000026
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [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
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [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
[1,4,3,2,5] => [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
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000382
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ 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%
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] => 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] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[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] => 1
[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] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[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] => 3
[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] => 3
[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] => 1
[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] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[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] => 1
[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] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[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] => 1
[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] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
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: St000505
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ 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},{2}}
=> 1
[2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [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},{2,4},{3}}
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[2,1,4,3] => [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]
=> {{1,3},{2},{4}}
=> 3
[2,3,4,1] => [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]
=> {{1,3,4},{2}}
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4
[3,4,1,2] => [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]
=> {{1,2,4},{3}}
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [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},{2,5},{3},{4}}
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,4,2,3,5] => [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},{2,5},{3,4}}
=> 1
[1,4,3,2,5] => [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},{2,5},{3,4}}
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000025
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000439
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 2 = 1 + 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
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 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 1
[1,2] => 1 => 1
[2,1] => 0 => 2
[1,2,3] => 11 => 1
[1,3,2] => 10 => 1
[2,1,3] => 01 => 2
[2,3,1] => 00 => 3
[3,1,2] => 00 => 3
[3,2,1] => 00 => 3
[1,2,3,4] => 111 => 1
[1,2,4,3] => 110 => 1
[1,3,2,4] => 101 => 1
[1,3,4,2] => 100 => 1
[1,4,2,3] => 100 => 1
[1,4,3,2] => 100 => 1
[2,1,3,4] => 011 => 2
[2,1,4,3] => 010 => 2
[2,3,1,4] => 001 => 3
[2,3,4,1] => 000 => 4
[2,4,1,3] => 000 => 4
[2,4,3,1] => 000 => 4
[3,1,2,4] => 001 => 3
[3,1,4,2] => 000 => 4
[3,2,1,4] => 001 => 3
[3,2,4,1] => 000 => 4
[3,4,1,2] => 000 => 4
[3,4,2,1] => 000 => 4
[4,1,2,3] => 000 => 4
[4,1,3,2] => 000 => 4
[4,2,1,3] => 000 => 4
[4,2,3,1] => 000 => 4
[4,3,1,2] => 000 => 4
[4,3,2,1] => 000 => 4
[1,2,3,4,5] => 1111 => 1
[1,2,3,5,4] => 1110 => 1
[1,2,4,3,5] => 1101 => 1
[1,2,4,5,3] => 1100 => 1
[1,2,5,3,4] => 1100 => 1
[1,2,5,4,3] => 1100 => 1
[1,3,2,4,5] => 1011 => 1
[1,3,2,5,4] => 1010 => 1
[1,3,4,2,5] => 1001 => 1
[1,3,4,5,2] => 1000 => 1
[1,3,5,2,4] => 1000 => 1
[1,3,5,4,2] => 1000 => 1
[1,4,2,3,5] => 1001 => 1
[1,4,2,5,3] => 1000 => 1
[1,4,3,2,5] => 1001 => 1
[1,4,3,5,2] => 1000 => 1
[1,4,5,2,3] => 1000 => 1
[1,4,5,3,2] => 1000 => 1
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
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? = 1 - 1
[1,2] => 1 => 0 => 0 = 1 - 1
[2,1] => 0 => 1 => 1 = 2 - 1
[1,2,3] => 11 => 00 => 0 = 1 - 1
[1,3,2] => 10 => 01 => 0 = 1 - 1
[2,1,3] => 01 => 10 => 1 = 2 - 1
[2,3,1] => 00 => 11 => 2 = 3 - 1
[3,1,2] => 00 => 11 => 2 = 3 - 1
[3,2,1] => 00 => 11 => 2 = 3 - 1
[1,2,3,4] => 111 => 000 => 0 = 1 - 1
[1,2,4,3] => 110 => 001 => 0 = 1 - 1
[1,3,2,4] => 101 => 010 => 0 = 1 - 1
[1,3,4,2] => 100 => 011 => 0 = 1 - 1
[1,4,2,3] => 100 => 011 => 0 = 1 - 1
[1,4,3,2] => 100 => 011 => 0 = 1 - 1
[2,1,3,4] => 011 => 100 => 1 = 2 - 1
[2,1,4,3] => 010 => 101 => 1 = 2 - 1
[2,3,1,4] => 001 => 110 => 2 = 3 - 1
[2,3,4,1] => 000 => 111 => 3 = 4 - 1
[2,4,1,3] => 000 => 111 => 3 = 4 - 1
[2,4,3,1] => 000 => 111 => 3 = 4 - 1
[3,1,2,4] => 001 => 110 => 2 = 3 - 1
[3,1,4,2] => 000 => 111 => 3 = 4 - 1
[3,2,1,4] => 001 => 110 => 2 = 3 - 1
[3,2,4,1] => 000 => 111 => 3 = 4 - 1
[3,4,1,2] => 000 => 111 => 3 = 4 - 1
[3,4,2,1] => 000 => 111 => 3 = 4 - 1
[4,1,2,3] => 000 => 111 => 3 = 4 - 1
[4,1,3,2] => 000 => 111 => 3 = 4 - 1
[4,2,1,3] => 000 => 111 => 3 = 4 - 1
[4,2,3,1] => 000 => 111 => 3 = 4 - 1
[4,3,1,2] => 000 => 111 => 3 = 4 - 1
[4,3,2,1] => 000 => 111 => 3 = 4 - 1
[1,2,3,4,5] => 1111 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 1110 => 0001 => 0 = 1 - 1
[1,2,4,3,5] => 1101 => 0010 => 0 = 1 - 1
[1,2,4,5,3] => 1100 => 0011 => 0 = 1 - 1
[1,2,5,3,4] => 1100 => 0011 => 0 = 1 - 1
[1,2,5,4,3] => 1100 => 0011 => 0 = 1 - 1
[1,3,2,4,5] => 1011 => 0100 => 0 = 1 - 1
[1,3,2,5,4] => 1010 => 0101 => 0 = 1 - 1
[1,3,4,2,5] => 1001 => 0110 => 0 = 1 - 1
[1,3,4,5,2] => 1000 => 0111 => 0 = 1 - 1
[1,3,5,2,4] => 1000 => 0111 => 0 = 1 - 1
[1,3,5,4,2] => 1000 => 0111 => 0 = 1 - 1
[1,4,2,3,5] => 1001 => 0110 => 0 = 1 - 1
[1,4,2,5,3] => 1000 => 0111 => 0 = 1 - 1
[1,4,3,2,5] => 1001 => 0110 => 0 = 1 - 1
[1,4,3,5,2] => 1000 => 0111 => 0 = 1 - 1
[1,4,5,2,3] => 1000 => 0111 => 0 = 1 - 1
[1,4,5,3,2] => 1000 => 0111 => 0 = 1 - 1
Description
The number of leading ones in a binary word.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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)
=> ? ∊ {1,1,1}
[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)
=> ? ∊ {1,1,1}
[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)
=> ? ∊ {1,1,1}
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.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000544The cop number of a graph. 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. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St001645The pebbling number of a connected graph. St000740The last entry of a permutation. St000501The size of the first part in the decomposition of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. 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. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000051The size of the left subtree of a binary tree. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the 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. St000141The maximum drop size of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000662The staircase size of the code of a permutation.
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