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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> 1
{{1,2}}
=> [2]
=> 0
{{1},{2}}
=> [1,1]
=> 2
{{1,2,3}}
=> [3]
=> 0
{{1,2},{3}}
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> 3
{{1,2,3,4}}
=> [4]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> 2
{{1,3,4},{2}}
=> [3,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> 2
{{1,4},{2,3}}
=> [2,2]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 2
{{1,4},{2},{3}}
=> [2,1,1]
=> 2
{{1},{2,4},{3}}
=> [2,1,1]
=> 2
{{1},{2},{3,4}}
=> [2,1,1]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 4
{{1,2,3,4,5}}
=> [5]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 2
{{1,2,4,5},{3}}
=> [4,1]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 2
{{1,2,5},{3,4}}
=> [3,2]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 3
{{1,3,4,5},{2}}
=> [4,1]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 2
{{1,3,5},{2,4}}
=> [3,2]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 3
{{1,4,5},{2,3}}
=> [3,2]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000445
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(load all 7 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001126
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
Description
Number of simple module that are 1-regular in the corresponding Nakayama algebra.
Matching statistic: St000439
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 1 + 2
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 4 + 2
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 0 + 2
Description
The position of the first down step of a Dyck path.
Matching statistic: St000247
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 1
{{1,2}}
=> 0
{{1},{2}}
=> 2
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 2
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 2
{{1,4},{2},{3}}
=> 2
{{1},{2,4},{3}}
=> 2
{{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> 4
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 0
{{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3,4}}
=> 0
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> 3
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 0
{{1,3,4},{2},{5}}
=> 2
{{1,3,5},{2,4}}
=> 0
{{1,3},{2,4,5}}
=> 0
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 3
{{1,4,5},{2,3}}
=> 0
{{1,4},{2,3,5}}
=> 0
{{1,4},{2,3},{5}}
=> 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 1
{{1,2}}
=> 2
{{1},{2}}
=> 0
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 2
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> 2
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 2
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 3
{{1,2,3,5},{4}}
=> 3
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> 3
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 3
{{1,2},{3,4,5}}
=> 3
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> 3
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 1
{{1,4},{2,3},{5}}
=> 1
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000674
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> ? = 1
{{1,2}}
=> [2] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000025
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00330: Dyck paths —rotate triangulation clockwise⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 100%●distinct values known / distinct values provided: 88%
Values
{{1}}
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3,4,5,6,7}}
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,7} + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,7} + 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000986
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1
{{1,2}}
=> [2] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> 2
{{1,2,3}}
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => ([(2,3)],4)
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> 4
{{1,2,3,4,5}}
=> [5] => [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)
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => ([(3,4)],5)
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,4},{3},{5,6,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,5},{3},{4,6,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,6},{3},{4,5,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,2,7},{3},{4,5,6}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,4},{2},{5,6,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,5},{2},{4,6,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,6},{2},{4,5,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,3,7},{2},{4,5,6}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4,5},{2},{3,6,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4,6},{2},{3,5,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,4,7},{2},{3,5,6}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5,6},{2},{3,4,7}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,5,7},{2},{3,4,6}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
{{1,6,7},{2},{3,4,5}}
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St000241
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 90%●distinct values known / distinct values provided: 88%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 90%●distinct values known / distinct values provided: 88%
Values
{{1}}
=> [1]
=> [1,0]
=> [1] => 1
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 2
{{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]
=> [1,3,2] => 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
{{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]
=> [1,2,4,3] => 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
{{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]
=> [1,2,3,5,4] => 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
{{1,2,3,4,5,6,7}}
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4,5,7},{6}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4,5},{6},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4,6},{5},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4,7},{5},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,5,6},{4},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,5,7},{4},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,6,7},{4},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,5,6},{3},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,5,7},{3},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,6,7},{3},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,5,6,7},{3},{4}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,6,7},{3},{4},{5}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2,7},{3},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,5,6},{2},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,5,7},{2},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,5},{2},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,6,7},{2},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,6},{2},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4,7},{2},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,4},{2},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,5,6,7},{2},{4}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,5,6},{2},{4},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,5,7},{2},{4},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,5},{2},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,6,7},{2},{4},{5}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,6},{2},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3,7},{2},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1,3},{2},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
{{1},{2,3,4,5,6},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7}
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000022The number of fixed points of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000215The number of adjacencies of a permutation, zero appended. St000895The number of ones on the main diagonal of an alternating sign matrix. St000221The number of strong fixed points of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St000894The trace of an alternating sign matrix. St001060The distinguishing index of a graph. St001903The number of fixed points of a parking function.
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