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Your data matches 163 different statistics following compositions of up to 3 maps.
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Matching statistic: St000026
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1] => 1 = 0 + 1
[1,1,0,0]
=> [2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [3] => 3 = 2 + 1
[1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [4] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 5 = 4 + 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1] => 1 = 0 + 1
[1,1,0,0]
=> [2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => 1 = 0 + 1
[1,1,0,1,0,0]
=> [3] => 3 = 2 + 1
[1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [4] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [4] => 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [4] => 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [4] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 5 = 4 + 1
Description
The last part of an integer composition.
Matching statistic: St000505
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
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%
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> 1 = 0 + 1
[1,0,1,0]
=> {{1},{2}}
=> 1 = 0 + 1
[1,1,0,0]
=> {{1,2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 5 = 4 + 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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%
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
[1,0]
=> [1] => [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[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
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[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
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[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
[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
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[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
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[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
[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
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[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
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 4
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: St000987
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 0
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St000234
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,2] => 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[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]
=> [1,2,3,4,5] => 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]
=> [2,1,3,4,5] => 3
[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]
=> [1,2,3,4,5] => 4
[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]
=> [3,1,2,4,5] => 2
[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]
=> [3,2,1,4,5] => 2
[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]
=> [1,2,3,4,5] => 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]
=> [2,1,3,4,5] => 3
[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]
=> [1,2,3,4,5] => 4
[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]
=> [4,1,2,3,5] => 1
[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]
=> [4,2,1,3,5] => 1
[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]
=> [1,2,3,4,5] => 4
[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]
=> [4,1,2,3,5] => 1
[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]
=> [4,3,1,2,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000319
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> [1]
=> 0
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 0
[1,1,1,0,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 3
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 3
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> [5]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> [1]
=> 0
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> [2]
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 2
[1,1,0,1,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 0
[1,1,1,0,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 3
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 3
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> [5]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001176
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => ([],1)
=> [1]
=> 0
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> [2]
=> 0
[1,1,0,0]
=> [2] => ([],2)
=> [1,1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
[1,1,0,1,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 2
[1,1,1,0,0,0]
=> [3] => ([],3)
=> [1,1,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> [1,1,1,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [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)
=> [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 4
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
The following 153 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000147The largest part of an integer partition. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000700The protection number of an ordered tree. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001389The number of partitions of the same length below the given integer partition. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000171The degree of the graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers 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. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St001725The harmonious chromatic number of a graph. St000019The cardinality of the support of a permutation. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001461The number of topologically connected components of the chord diagram of a permutation. St000653The last descent of a permutation. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000883The number of longest increasing subsequences of a permutation. St000617The number of global maxima of a Dyck path. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000237The number of small exceedances. St000225Difference between largest and smallest parts in a partition. St000740The last entry of a permutation. St001316The domatic number of a graph. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000501The size of the first part in the decomposition of a permutation. St000795The mad of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001645The pebbling number of a connected graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001175The size of a partition minus the hook length of the base cell. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001657The number of twos in an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001497The position of the largest weak excedence of a permutation. St001330The hat guessing number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. 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. St001481The minimal height of a peak of a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000030The sum of the descent differences of a permutations. St000090The variation of a composition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000553The number of blocks of a graph. St000990The first ascent of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000770The major index of an integer partition when read from bottom to top. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001394The genus of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000260The radius of a connected graph. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000060The greater neighbor of the maximum. 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. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001413Half the length of the longest even length palindromic prefix of a binary word. St001434The number of negative sum pairs of a signed permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001557The number of inversions of the second entry of a permutation. St000005The bounce statistic of a Dyck path. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000338The number of pixed points of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
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