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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St000442
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Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000013
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Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000521
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [[],[]]
=> 2 = 0 + 2
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [[[]]]
=> 3 = 1 + 2
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 2 = 0 + 2
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 3 = 1 + 2
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 3 = 1 + 2
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 3 = 1 + 2
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 3 = 1 + 2
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2 = 0 + 2
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 1 + 2
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 1 + 2
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 4 = 2 + 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 1 + 2
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 4 = 2 + 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 4 = 2 + 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 1 + 2
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3 = 1 + 2
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2 = 0 + 2
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 1 + 2
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 1 + 2
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 1 + 2
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 1 + 2
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 1 + 2
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 1 + 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 3 = 1 + 2
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3 = 1 + 2
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 1 + 2
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 1 + 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 3 = 1 + 2
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
Description
The number of distinct subtrees of an ordered tree.
A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree $[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.
Matching statistic: St001039
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000676
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000024
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St001007
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000053
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000211
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> {{1,2}}
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [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
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,8},{4},{5},{6},{7}}
=> ? = 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 1
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St001197
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,8},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6},{7},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,8},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4,8},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4,6,7,8},{5}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2},{3},{4,5,6,7,8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2},{3,5,7,8},{4},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2},{3,5,6,7,8},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,8},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,8},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2},{3,4,5,6,7,8}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
{{1},{2,8},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2,5,6,8},{3},{4},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,5,6,7,8},{3},{4}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,4,6,7},{3},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,4,6,8},{3},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,4,6,7,8},{3},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,4,5,7,8},{3},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,4,5,6,8},{3},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,4,5,6,7,8},{3}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2,3,5},{4},{6},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2,3,6},{4},{5},{7},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2,3,7},{4},{5},{6},{8}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2,3,8},{4},{5},{6},{7}}
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1},{2,3,7,8},{4},{5},{6}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,3,6,7},{4},{5},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,3,6,8},{4},{5},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,3,6,7,8},{4},{5}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,7},{4},{6},{8}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,3,5,8},{4},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,3,5,7,8},{4},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,6,7},{4},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,3,5,6,8},{4},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,8},{5},{6},{7}}
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1},{2,3,4,7,8},{5},{6}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,6,8},{5},{7}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,6,7,8},{5}}
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
{{1},{2,3,4,5,6},{7},{8}}
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000662The staircase size of the code of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000306The bounce count of a Dyck path. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000702The number of weak deficiencies of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000325The width of the tree associated to a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St000094The depth of an ordered tree. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001589The nesting number of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral.
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