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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
(load all 17 compositions to match this statistic)
(load all 17 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
(load all 14 compositions to match this statistic)
(load all 14 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: St000247
(load all 269 compositions to match this statistic)
(load all 269 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
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00221: Set partitions —conjugate⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> ? = 1
{{1,2}}
=> {{1},{2}}
=> 0
{{1},{2}}
=> {{1,2}}
=> 2
{{1,2,3}}
=> {{1},{2},{3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1},{2,3}}
=> {{1,3},{2}}
=> 1
{{1},{2},{3}}
=> {{1,2,3}}
=> 3
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 0
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 2
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 0
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 2
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 2
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 4
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> 0
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> 0
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> 0
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> 2
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> 0
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> 0
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 2
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> 3
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> 0
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> 0
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> 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 16 compositions to match this statistic)
(load all 16 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: 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
{{1,2,4},{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
{{1,2,5},{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
{{1,2,6},{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
{{1,2,7},{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
{{1,3,4},{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
{{1,3,5},{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
{{1,3,6},{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
{{1,3,7},{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
{{1,4,5},{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
{{1,4,6},{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
{{1,4,7},{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
{{1,5,6},{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
{{1,5,7},{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
{{1,6,7},{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
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St000241
(load all 26 compositions to match this statistic)
(load all 26 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
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ? = 1
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 4
{{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] => ? = 5
{{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] => ? = 1
{{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] => ? = 2
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 2
{{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] => ? = 3
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 3
{{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] => ? = 4
{{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] => ? = 4
{{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] => ? = 5
{{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] => ? = 1
{{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] => ? = 2
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000022
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1] => 1
{{1,2}}
=> [2] => [1,1,0,0]
=> [2,1] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,2] => 2
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [3,1,2] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 0
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 0
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 0
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 0
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 1
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 2
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 2
{{1,2,4,5},{3,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,4,5},{3,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,4,6},{3,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,4,6},{3,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,4,7},{3,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,4},{3,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 0
{{1,2,4},{3,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
{{1,2,4,7},{3,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,4},{3,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
{{1,2,4},{3,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 0
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2
{{1,2,4,6},{3,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,4},{3,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
{{1,2,4},{3,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 0
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2
{{1,2,4},{3,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 0
{{1,2,4},{3},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 1
{{1,2,4},{3},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 2
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2
{{1,2,4},{3},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 2
{{1,2,5,6},{3,4,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
{{1,2,5,6},{3,4},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1
{{1,2,5,7},{3,4,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 0
Description
The number of fixed points of a permutation.
Matching statistic: St000895
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 88%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 88%
Values
{{1}}
=> [1] => [1,0]
=> [[1]]
=> 1
{{1,2}}
=> [2] => [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
{{1,2,3,4,5,6,7}}
=> [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3,4,5},{6},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,4,6},{5},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,4},{5,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3,4},{5,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,4,7},{5},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,4},{5,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,4},{5},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
{{1,2,3,5,6},{4},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,5},{4,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3,5},{4,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,5,7},{4},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,5},{4,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,5},{4},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
{{1,2,3,6},{4,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3,6},{4,5},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,7},{4,5,6}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3},{4,5,6,7}}
=> [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3},{4,5,6},{7}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,7},{4,5},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3},{4,5,7},{6}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,6},{4,7},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3,6},{4},{5,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3},{4,6,7},{5}}
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,7},{4},{5,6}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,3},{4},{5,6,7}}
=> [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 2
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 4
{{1,2,4,5,6},{3},{7}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,4,5},{3,6,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
{{1,2,4,5},{3,6},{7}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,4,5,7},{3},{6}}
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
{{1,2,4,5},{3,7},{6}}
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
{{1,2,4,5},{3},{6,7}}
=> [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
{{1,2,4,6},{3,5,7}}
=> [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 0
Description
The number of ones on the main diagonal of an alternating sign matrix.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001903The number of fixed points of a parking function.
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