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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000250
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St000250: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 3
{{1},{2}}
=> 2
{{1,2,3}}
=> 4
{{1,2},{3}}
=> 3
{{1,3},{2}}
=> 3
{{1},{2,3}}
=> 3
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 5
{{1,2,3},{4}}
=> 4
{{1,2,4},{3}}
=> 4
{{1,2},{3,4}}
=> 4
{{1,2},{3},{4}}
=> 4
{{1,3,4},{2}}
=> 4
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 3
{{1,4},{2,3}}
=> 4
{{1},{2,3,4}}
=> 4
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 4
{{1},{2,4},{3}}
=> 3
{{1},{2},{3,4}}
=> 4
{{1},{2},{3},{4}}
=> 4
{{1,2,3,4,5}}
=> 6
{{1,2,3,4},{5}}
=> 5
{{1,2,3,5},{4}}
=> 5
{{1,2,3},{4,5}}
=> 5
{{1,2,3},{4},{5}}
=> 5
{{1,2,4,5},{3}}
=> 5
{{1,2,4},{3,5}}
=> 3
{{1,2,4},{3},{5}}
=> 4
{{1,2,5},{3,4}}
=> 5
{{1,2},{3,4,5}}
=> 5
{{1,2},{3,4},{5}}
=> 5
{{1,2,5},{3},{4}}
=> 5
{{1,2},{3,5},{4}}
=> 4
{{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> 5
{{1,3,4,5},{2}}
=> 5
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 4
{{1,3,5},{2,4}}
=> 3
{{1,3},{2,4,5}}
=> 3
{{1,3},{2,4},{5}}
=> 3
{{1,3,5},{2},{4}}
=> 4
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 4
{{1,3},{2},{4},{5}}
=> 4
{{1,4,5},{2,3}}
=> 5
{{1,4},{2,3,5}}
=> 3
{{1,4},{2,3},{5}}
=> 4
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 1 = 3 - 2
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 0 = 2 - 2
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 2 = 4 - 2
{{1,2},{3}}
=> [2,1,3] => [3,2,1] => [2,3,1] => 1 = 3 - 2
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1 = 3 - 2
{{1},{2,3}}
=> [1,3,2] => [2,1,3] => [2,1,3] => 1 = 3 - 2
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 1 = 3 - 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
{{1,2,3},{4}}
=> [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 2 = 4 - 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 2 = 4 - 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 2 = 4 - 2
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0 = 2 - 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1 = 3 - 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 2 = 4 - 2
{{1},{2,3,4}}
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 2 = 4 - 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 2 = 4 - 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1 = 3 - 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2 = 4 - 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 2 = 4 - 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 6 - 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 3 = 5 - 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 5 - 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 3 = 5 - 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,1,4] => 3 = 5 - 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 5 - 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,3,4] => [2,5,4,3,1] => 1 = 3 - 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 2 = 4 - 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 5 - 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 3 = 5 - 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,1,3] => 3 = 5 - 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 5 - 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 4 - 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2,4,1,5] => [2,4,1,3,5] => 3 = 5 - 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 3 = 5 - 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 3 = 5 - 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,2,4,3] => [4,5,3,2,1] => 1 = 3 - 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,3,2,4,1] => [3,2,4,5,1] => 2 = 4 - 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1 = 3 - 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,1,2,3,5] => [4,3,2,1,5] => 1 = 3 - 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,5,2,3,1] => [5,1,4,3,2] => 1 = 3 - 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 4 - 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,1,2,5,3] => [5,3,2,1,4] => 1 = 3 - 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,3,2,1,5] => [3,2,4,1,5] => 2 = 4 - 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [4,3,2,5,1] => [3,2,5,1,4] => 2 = 4 - 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,3,2,5] => [1,3,4,2,5] => 3 = 5 - 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,1,3,2,4] => [3,5,4,2,1] => 1 = 3 - 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,4,3,2,1] => [3,4,2,5,1] => 2 = 4 - 2
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [7,2,3,4,1,5,6] => [2,3,4,7,6,5,1] => ? = 5 - 2
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [7,2,3,4,6,5,1] => [2,3,4,6,5,7,1] => ? = 6 - 2
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [5,2,3,4,6,1,7] => [2,3,4,6,1,5,7] => ? = 7 - 2
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [7,2,3,1,4,6,5] => [2,3,6,7,5,4,1] => ? = 5 - 2
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [7,2,3,5,4,6,1] => [2,3,5,4,6,7,1] => ? = 6 - 2
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [6,2,3,1,4,5,7] => [2,3,6,5,4,1,7] => ? = 5 - 2
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [6,2,3,7,4,5,1] => [2,3,7,1,6,5,4] => ? = 5 - 2
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [6,2,3,1,4,7,5] => [2,3,7,5,4,1,6] => ? = 5 - 2
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [6,2,3,5,4,1,7] => [2,3,5,4,6,1,7] => ? = 6 - 2
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [6,2,3,5,4,7,1] => [2,3,5,4,7,1,6] => ? = 6 - 2
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [7,2,3,1,5,4,6] => [2,3,5,7,6,4,1] => ? = 5 - 2
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => ? = 6 - 2
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [4,2,3,6,5,1,7] => [2,3,5,6,1,4,7] => ? = 7 - 2
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [4,2,3,6,5,7,1] => [2,3,5,7,1,4,6] => ? = 7 - 2
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [7,2,3,1,6,4,5] => [2,3,7,5,6,4,1] => ? = 5 - 2
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [7,2,3,5,1,4,6] => [2,3,7,6,4,5,1] => ? = 5 - 2
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [7,2,3,5,6,4,1] => [2,3,6,4,5,7,1] => ? = 6 - 2
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [4,2,3,1,6,5,7] => [2,3,4,1,6,5,7] => ? = 6 - 2
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [4,2,3,7,1,5,6] => [2,3,7,6,5,1,4] => ? = 5 - 2
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [4,2,3,7,6,5,1] => [2,3,6,5,7,1,4] => ? = 6 - 2
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [4,2,3,1,7,6,5] => [2,3,4,1,6,7,5] => ? = 6 - 2
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [4,2,3,5,1,6,7] => [2,3,5,1,4,6,7] => ? = 7 - 2
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [4,2,3,5,7,6,1] => [2,3,6,7,1,4,5] => ? = 7 - 2
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [4,2,3,1,6,7,5] => [2,3,4,1,7,5,6] => ? = 6 - 2
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [4,2,3,5,1,7,6] => [2,3,5,1,4,7,6] => ? = 6 - 2
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => [4,2,3,5,6,1,7] => [2,3,6,1,4,5,7] => ? = 7 - 2
{{1,2,3},{4},{5},{6},{7}}
=> [2,3,1,4,5,6,7] => [4,2,3,5,6,7,1] => [2,3,7,1,4,5,6] => ? = 7 - 2
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [7,2,1,3,5,6,4] => [2,5,6,7,4,3,1] => ? = 5 - 2
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [7,2,4,3,5,6,1] => [2,4,3,5,6,7,1] => ? = 6 - 2
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [6,2,1,3,5,4,7] => [2,5,6,4,3,1,7] => ? = 5 - 2
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [6,2,7,3,5,4,1] => [2,5,7,1,6,4,3] => ? = 5 - 2
{{1,2,4,5},{3,7},{6}}
=> [2,4,7,5,1,6,3] => [6,2,1,3,5,7,4] => [2,5,7,4,3,1,6] => ? = 5 - 2
{{1,2,4,5},{3},{6,7}}
=> [2,4,3,5,1,7,6] => [6,2,4,3,5,1,7] => [2,4,3,5,6,1,7] => ? = 6 - 2
{{1,2,4,5},{3},{6},{7}}
=> [2,4,3,5,1,6,7] => [6,2,4,3,5,7,1] => [2,4,3,5,7,1,6] => ? = 6 - 2
{{1,2,4,6},{3,5,7}}
=> [2,4,5,6,7,1,3] => [7,2,1,3,4,5,6] => [2,7,6,5,4,3,1] => ? = 3 - 2
{{1,2,4,6},{3,5},{7}}
=> [2,4,5,6,3,1,7] => [7,2,6,3,4,5,1] => [2,6,5,4,3,7,1] => ? = 4 - 2
{{1,2,4},{3,5,6,7}}
=> [2,4,5,1,6,7,3] => [5,2,1,3,4,6,7] => [2,5,4,3,1,6,7] => ? = 5 - 2
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [5,2,7,3,4,6,1] => [2,6,7,1,5,4,3] => ? = 5 - 2
{{1,2,4},{3,5,7},{6}}
=> [2,4,5,1,7,6,3] => [5,2,1,3,4,7,6] => [2,5,4,3,1,7,6] => ? = 4 - 2
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [5,2,6,3,4,1,7] => [2,6,1,5,4,3,7] => ? = 5 - 2
{{1,2,4},{3,5},{6},{7}}
=> [2,4,5,1,3,6,7] => [5,2,6,3,4,7,1] => [2,7,1,5,4,3,6] => ? = 5 - 2
{{1,2,4,6},{3,7},{5}}
=> [2,4,7,6,5,1,3] => [7,2,1,3,6,5,4] => [2,6,5,7,4,3,1] => ? = 4 - 2
{{1,2,4,6},{3},{5,7}}
=> [2,4,3,6,7,1,5] => [7,2,4,3,1,5,6] => [2,4,3,7,6,5,1] => ? = 4 - 2
{{1,2,4,6},{3},{5},{7}}
=> [2,4,3,6,5,1,7] => [7,2,4,3,6,5,1] => [2,4,3,6,5,7,1] => ? = 5 - 2
{{1,2,4},{3,6,7},{5}}
=> [2,4,6,1,5,7,3] => [5,2,1,3,6,4,7] => [2,6,4,3,1,5,7] => ? = 5 - 2
{{1,2,4},{3,6},{5,7}}
=> [2,4,6,1,7,3,5] => [5,2,7,3,1,4,6] => [2,5,1,7,6,4,3] => ? = 4 - 2
{{1,2,4},{3,6},{5},{7}}
=> [2,4,6,1,5,3,7] => [5,2,7,3,6,4,1] => [2,7,1,5,6,4,3] => ? = 5 - 2
{{1,2,4},{3,7},{5,6}}
=> [2,4,7,1,6,5,3] => [5,2,1,3,7,6,4] => [2,6,7,4,3,1,5] => ? = 5 - 2
{{1,2,4},{3},{5,6,7}}
=> [2,4,3,1,6,7,5] => [5,2,4,3,1,6,7] => [2,4,3,5,1,6,7] => ? = 6 - 2
{{1,2,4},{3},{5,6},{7}}
=> [2,4,3,1,6,5,7] => [5,2,4,3,7,6,1] => [2,4,3,6,7,1,5] => ? = 6 - 2
Description
The number of ascents of a permutation.
Matching statistic: St000702
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 88%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 88%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 2 = 3 - 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 3 - 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 2 = 3 - 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [3,2,1] => 2 = 3 - 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3 = 4 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 3 = 4 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3 = 4 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 4 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3 = 4 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [3,1,2,4] => 3 = 4 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 3 = 4 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4 = 5 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 4 = 5 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 3 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4 = 5 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4 = 5 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,5,3,2,4] => 4 = 5 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,4,2,3,5] => 4 = 5 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,5,2,4,3] => 4 = 5 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 4 = 5 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 4 = 5 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [2,4,3,5,1] => 2 = 3 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 4 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [2,3,4,1,5] => 2 = 3 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [2,3,5,4,1] => 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [2,3,5,1,4] => 2 = 3 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [2,4,5,3,1] => 2 = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 3 = 4 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,5,3,4] => 3 = 4 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => 4 = 5 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [3,2,4,5,1] => 2 = 3 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 4 - 1
{{1,2,6,7},{3,4,5}}
=> [2,6,4,5,3,7,1] => [1,5,3,4,2,6,7] => [1,5,3,4,2,6,7] => ? = 7 - 1
{{1,2,6},{3,4,5,7}}
=> [2,6,4,5,7,1,3] => [1,7,3,4,2,5,6] => [1,5,3,4,6,7,2] => ? = 5 - 1
{{1,2,6},{3,4,5},{7}}
=> [2,6,4,5,3,1,7] => [1,5,3,4,2,7,6] => [1,5,3,4,2,7,6] => ? = 6 - 1
{{1,2,7},{3,4,5,6}}
=> [2,7,4,5,6,3,1] => [1,6,3,4,5,2,7] => [1,6,3,4,5,2,7] => ? = 7 - 1
{{1,2},{3,4,5,6,7}}
=> [2,1,4,5,6,7,3] => [1,7,3,4,5,6,2] => [1,7,3,4,5,6,2] => ? = 7 - 1
{{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => [1,6,3,4,5,7,2] => [1,7,3,4,5,2,6] => ? = 7 - 1
{{1,2,7},{3,4,5},{6}}
=> [2,7,4,5,3,6,1] => [1,5,3,4,6,2,7] => [1,6,3,4,2,5,7] => ? = 7 - 1
{{1,2},{3,4,5,7},{6}}
=> [2,1,4,5,7,6,3] => [1,7,3,4,6,5,2] => [1,7,3,4,6,5,2] => ? = 6 - 1
{{1,2},{3,4,5},{6,7}}
=> [2,1,4,5,3,7,6] => [1,5,3,4,7,6,2] => [1,7,3,4,2,6,5] => ? = 7 - 1
{{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => [1,5,3,4,6,7,2] => [1,7,3,4,2,5,6] => ? = 7 - 1
{{1,2,6,7},{3,4},{5}}
=> [2,6,4,3,5,7,1] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 7 - 1
{{1,2,6},{3,4,7},{5}}
=> [2,6,4,7,5,1,3] => [1,7,3,5,2,4,6] => [1,5,3,6,4,7,2] => ? = 5 - 1
{{1,2,6},{3,4},{5,7}}
=> [2,6,4,3,7,1,5] => [1,4,3,7,2,5,6] => [1,5,3,2,6,7,4] => ? = 5 - 1
{{1,2,6},{3,4},{5},{7}}
=> [2,6,4,3,5,1,7] => [1,4,3,5,2,7,6] => [1,5,3,2,4,7,6] => ? = 6 - 1
{{1,2,7},{3,4,6},{5}}
=> [2,7,4,6,5,3,1] => [1,6,3,5,4,2,7] => [1,6,3,5,4,2,7] => ? = 6 - 1
{{1,2},{3,4,6,7},{5}}
=> [2,1,4,6,5,7,3] => [1,7,3,5,4,6,2] => [1,7,3,5,4,6,2] => ? = 6 - 1
{{1,2},{3,4,6},{5,7}}
=> [2,1,4,6,7,3,5] => [1,6,3,7,4,5,2] => [1,7,3,5,6,2,4] => ? = 5 - 1
{{1,2},{3,4,6},{5},{7}}
=> [2,1,4,6,5,3,7] => [1,6,3,5,4,7,2] => [1,7,3,5,4,2,6] => ? = 6 - 1
{{1,2,7},{3,4},{5,6}}
=> [2,7,4,3,6,5,1] => [1,4,3,6,5,2,7] => [1,6,3,2,5,4,7] => ? = 7 - 1
{{1,2},{3,4,7},{5,6}}
=> [2,1,4,7,6,5,3] => [1,7,3,6,5,4,2] => [1,7,3,6,5,4,2] => ? = 6 - 1
{{1,2},{3,4},{5,6,7}}
=> [2,1,4,3,6,7,5] => [1,4,3,7,5,6,2] => [1,7,3,2,5,6,4] => ? = 7 - 1
{{1,2},{3,4},{5,6},{7}}
=> [2,1,4,3,6,5,7] => [1,4,3,6,5,7,2] => [1,7,3,2,5,4,6] => ? = 7 - 1
{{1,2,7},{3,4},{5},{6}}
=> [2,7,4,3,5,6,1] => [1,4,3,5,6,2,7] => [1,6,3,2,4,5,7] => ? = 7 - 1
{{1,2},{3,4,7},{5},{6}}
=> [2,1,4,7,5,6,3] => [1,7,3,5,6,4,2] => [1,7,3,6,4,5,2] => ? = 6 - 1
{{1,2},{3,4},{5,7},{6}}
=> [2,1,4,3,7,6,5] => [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => ? = 6 - 1
{{1,2},{3,4},{5},{6,7}}
=> [2,1,4,3,5,7,6] => [1,4,3,5,7,6,2] => [1,7,3,2,4,6,5] => ? = 7 - 1
{{1,2},{3,4},{5},{6},{7}}
=> [2,1,4,3,5,6,7] => [1,4,3,5,6,7,2] => [1,7,3,2,4,5,6] => ? = 7 - 1
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [1,7,6,2,4,3,5] => [1,4,6,5,7,3,2] => ? = 4 - 1
{{1,2,6,7},{3,5},{4}}
=> [2,6,5,4,3,7,1] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 6 - 1
{{1,2,6},{3,5,7},{4}}
=> [2,6,5,4,7,1,3] => [1,7,4,3,2,5,6] => [1,5,4,3,6,7,2] => ? = 4 - 1
{{1,2,6},{3,5},{4,7}}
=> [2,6,5,7,3,1,4] => [1,5,7,3,2,4,6] => [1,5,4,6,2,7,3] => ? = 4 - 1
{{1,2,6},{3,5},{4},{7}}
=> [2,6,5,4,3,1,7] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 5 - 1
{{1,2,7},{3,5,6},{4}}
=> [2,7,5,4,6,3,1] => [1,6,4,3,5,2,7] => [1,6,4,3,5,2,7] => ? = 6 - 1
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [1,7,4,3,5,6,2] => [1,7,4,3,5,6,2] => ? = 6 - 1
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [1,6,7,3,5,4,2] => [1,7,4,6,5,2,3] => ? = 5 - 1
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [1,6,4,3,5,7,2] => [1,7,4,3,5,2,6] => ? = 6 - 1
{{1,2,7},{3,5},{4,6}}
=> [2,7,5,6,3,4,1] => [1,5,6,3,4,2,7] => [1,6,4,5,2,3,7] => ? = 5 - 1
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [1,7,6,3,4,5,2] => [1,7,4,5,6,3,2] => ? = 4 - 1
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [1,5,7,3,4,6,2] => [1,7,4,5,2,6,3] => ? = 5 - 1
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [1,5,6,3,4,7,2] => [1,7,4,5,2,3,6] => ? = 5 - 1
{{1,2,7},{3,5},{4},{6}}
=> [2,7,5,4,3,6,1] => [1,5,4,3,6,2,7] => [1,6,4,3,2,5,7] => ? = 6 - 1
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [1,7,4,3,6,5,2] => [1,7,4,3,6,5,2] => ? = 5 - 1
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [1,5,7,3,6,4,2] => [1,7,4,6,2,5,3] => ? = 5 - 1
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [1,5,4,3,7,6,2] => [1,7,4,3,2,6,5] => ? = 6 - 1
{{1,2},{3,5},{4},{6},{7}}
=> [2,1,5,4,3,6,7] => [1,5,4,3,6,7,2] => [1,7,4,3,2,5,6] => ? = 6 - 1
{{1,2,6,7},{3},{4,5}}
=> [2,6,3,5,4,7,1] => [1,3,5,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 7 - 1
{{1,2,6},{3,7},{4,5}}
=> [2,6,7,5,4,1,3] => [1,7,5,4,2,3,6] => [1,5,6,4,3,7,2] => ? = 5 - 1
{{1,2,6},{3},{4,5,7}}
=> [2,6,3,5,7,1,4] => [1,3,7,4,2,5,6] => [1,5,2,4,6,7,3] => ? = 5 - 1
{{1,2,6},{3},{4,5},{7}}
=> [2,6,3,5,4,1,7] => [1,3,5,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 6 - 1
{{1,2,7},{3,6},{4,5}}
=> [2,7,6,5,4,3,1] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 6 - 1
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000213
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Values
{{1,2}}
=> [2,1] => [1,2] => 2 = 3 - 1
{{1},{2}}
=> [1,2] => [2,1] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 3 = 4 - 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => 2 = 3 - 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => 2 = 3 - 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => 2 = 3 - 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 4 = 5 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => 3 = 4 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => 3 = 4 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => 3 = 4 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => 3 = 4 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => 3 = 4 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => 2 = 3 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => 3 = 4 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => 3 = 4 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => 3 = 4 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => 3 = 4 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => 2 = 3 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => 3 = 4 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 6 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => 4 = 5 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => 4 = 5 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => 4 = 5 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => 4 = 5 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => 4 = 5 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => 2 = 3 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => 3 = 4 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => 4 = 5 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => 4 = 5 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => 4 = 5 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => 4 = 5 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => 3 = 4 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => 4 = 5 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => 4 = 5 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => 4 = 5 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => 2 = 3 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => 3 = 4 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => 2 = 3 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => 2 = 3 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => 3 = 4 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => 2 = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => 3 = 4 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => 3 = 4 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => 4 = 5 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => 2 = 3 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => 3 = 4 - 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 8 - 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,2,3,4,5,7,6] => ? = 7 - 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 7 - 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => ? = 7 - 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => ? = 7 - 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 7 - 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,2,3,7,4,5,6] => ? = 5 - 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,2,3,5,4,7,6] => ? = 6 - 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 7 - 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,2,3,7,5,6,4] => ? = 7 - 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,2,3,6,5,7,4] => ? = 7 - 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 7 - 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,2,3,7,6,5,4] => ? = 6 - 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,2,3,5,7,6,4] => ? = 7 - 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,2,3,5,6,7,4] => ? = 7 - 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ? = 7 - 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,2,7,3,5,4,6] => ? = 5 - 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,2,4,3,5,7,6] => ? = 6 - 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 5 - 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,2,7,3,4,6,5] => ? = 5 - 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,2,6,3,4,7,5] => ? = 5 - 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ? = 6 - 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,2,7,3,6,4,5] => ? = 5 - 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,2,4,3,7,6,5] => ? = 6 - 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,2,4,3,6,7,5] => ? = 6 - 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ? = 7 - 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,2,7,4,3,5,6] => ? = 5 - 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,2,5,4,3,7,6] => ? = 6 - 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,2,6,4,5,3,7] => ? = 7 - 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,2,7,4,5,6,3] => ? = 7 - 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,2,6,4,5,7,3] => ? = 7 - 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,2,5,4,6,3,7] => ? = 7 - 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,2,7,4,6,5,3] => ? = 6 - 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,2,5,4,7,6,3] => ? = 7 - 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,2,5,4,6,7,3] => ? = 7 - 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ? = 7 - 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,2,7,5,3,4,6] => ? = 5 - 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,2,4,7,3,5,6] => ? = 5 - 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,2,4,5,3,7,6] => ? = 6 - 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,6,5,4,3,7] => ? = 6 - 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,2,7,5,4,6,3] => ? = 6 - 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,2,6,7,4,5,3] => ? = 5 - 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,2,6,5,4,7,3] => ? = 6 - 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,2,4,6,5,3,7] => ? = 7 - 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,2,7,6,5,4,3] => ? = 6 - 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,2,4,7,5,6,3] => ? = 7 - 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,2,4,6,5,7,3] => ? = 7 - 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => ? = 7 - 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,2,7,5,6,4,3] => ? = 6 - 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,2,4,7,6,5,3] => ? = 6 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St001863
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 62%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 62%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 2 = 3 - 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1 = 2 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 4 - 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 3 - 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 2 = 3 - 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [3,2,1] => 2 = 3 - 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [2,3,1] => 2 = 3 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3 = 4 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 3 = 4 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3 = 4 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 4 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 3 = 4 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 3 = 4 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 3 = 4 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 3 = 4 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 6 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 5 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 5 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4 = 5 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 4 = 5 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 2 = 3 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4 = 5 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4 = 5 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 4 = 5 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 4 = 5 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 4 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 4 = 5 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 4 = 5 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 5 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 3 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 4 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 3 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 3 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 4 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 4 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 4 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 5 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 3 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 4 - 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 5 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 5 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 5 - 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 5 - 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 4 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 5 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 5 - 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 5 - 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 3 - 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 3 - 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 4 - 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 4 - 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 4 - 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3 - 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 4 - 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 5 - 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 4 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 5 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 5 - 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 5 - 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 4 - 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 4 - 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 5 - 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 5 - 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 7 - 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 6 - 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 6 - 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 6 - 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 6 - 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 6 - 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 4 - 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 5 - 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 6 - 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 6 - 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,5,4,6,3] => ? = 6 - 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ? = 6 - 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 5 - 1
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
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