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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000293
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => 1 => 0
{{1},{2}}
=> [1,2] => [1,2] => 0 => 0
{{1,2,3}}
=> [2,3,1] => [3,2,1] => 11 => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 10 => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 01 => 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 01 => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 00 => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,3,2,1] => 111 => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => 110 => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,2,1] => 101 => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 101 => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 100 => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,3,1] => 011 => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,3,2] => 011 => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 010 => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 011 => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => 011 => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 010 => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 001 => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 001 => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 001 => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,4,3,2,1] => 1111 => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,1,5] => 1110 => 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,3,2,1] => 1101 => 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => 1101 => 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => 1100 => 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,4,2,1] => 1011 => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,4,3] => 1011 => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,2,1,5] => 1010 => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,2,1] => 1011 => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,2,1] => 1001 => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 1001 => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,4,3,1] => 0111 => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,5,3,2] => 0111 => 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,3,1,5] => 0110 => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,2,4,3,1] => 0111 => 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,4,1,3,2] => 0111 => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,3,2,5] => 0110 => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,3,1] => 0101 => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,1,3,2] => 0101 => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 0101 => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 0100 => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,4,1] => 0111 => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,3,1,4,2] => 0111 => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => 0110 => 2
Description
The number of inversions of a binary word.
Matching statistic: St000589
(load all 218 compositions to match this statistic)
(load all 218 compositions to match this statistic)
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 58%
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 58%
Values
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
{{1,3},{2}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 0
{{1},{2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 2
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 1
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 2
{{1,3,4},{2}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1,2,4},{3}}
=> 0
{{1,3},{2,4}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 0
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1,3},{2,4}}
=> 0
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 0
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 3
{{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 2
{{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 2
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 4
{{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 3
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 1
{{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,3},{2,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 3
{{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 2
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 2
{{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 2
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 3
{{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> {{1,2,3,5},{4}}
=> 0
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> {{1,4},{2,3,5}}
=> 0
{{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,5},{4}}
=> 2
{{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 0
{{1,3},{2,4,5}}
=> {{1,5},{2,3,4}}
=> {{1,3,4},{2,5}}
=> {{1,2,4},{3,5}}
=> 0
{{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2,5},{4}}
=> {{1,3,5},{2},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 2
{{1,4,5},{2,3}}
=> {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 0
{{1,4},{2,3,5}}
=> {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> {{1,3,4},{2,5}}
=> 0
{{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 2
{{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> {{1,5,7,8},{2},{3,6},{4}}
=> ? = 6
{{1,3},{2,4},{5,6},{7,8}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> {{1,3,4},{2,6,8},{5},{7}}
=> {{1,4},{2},{3,5,7},{6,8}}
=> ? = 5
{{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> {{1,5,6,7},{2},{3,8},{4}}
=> ? = 5
{{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,8},{2,6},{4,5},{7}}
=> {{1,5},{2,3,8},{4,6},{7}}
=> {{1,5},{2},{3,4,8},{6,7}}
=> ? = 4
{{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> {{1,3,8},{2},{4,5,6},{7}}
=> {{1,7},{2},{3,4,5},{6,8}}
=> ? = 4
{{1,7},{2,3},{4,5},{6,8}}
=> {{1,3,5},{2,8},{4},{6,7}}
=> {{1,5},{2,3,7},{4},{6,8}}
=> {{1,2,5},{3},{4,8},{6,7}}
=> ? = 3
{{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> {{1,3,5,8},{2},{4},{6,7}}
=> {{1,6,8},{2,3},{4,7},{5}}
=> ? = 3
{{1,8},{2,4},{3,5},{6,7}}
=> {{1,7},{2,4,5},{3},{6,8}}
=> {{1,4,5},{2,8},{3},{6,7}}
=> {{1,5,8},{2,3},{4,7},{6}}
=> ? = 2
{{1,7},{2,4},{3,5},{6,8}}
=> {{1},{2,4,5,8},{3},{6,7}}
=> {{1},{2,4,5,7},{3},{6,8}}
=> {{1,2,5,7},{3},{4,6},{8}}
=> ? = 2
{{1,6},{2,4},{3,5},{7,8}}
=> {{1,8},{2,4,5},{3,6},{7}}
=> {{1,5},{2,4,6},{3,8},{7}}
=> {{1,3,5,7},{2},{4,8},{6}}
=> ? = 3
{{1,5},{2,4},{3,6},{7,8}}
=> {{1,8},{2,4},{3,5,6},{7}}
=> {{1,5,6},{2,4},{3,8},{7}}
=> {{1,3,6},{2},{4,8},{5,7}}
=> ? = 3
{{1,4},{2,5},{3,6},{7,8}}
=> {{1,8},{2},{3,4,5,6},{7}}
=> {{1,4,5,6},{2},{3,8},{7}}
=> {{1,3,6},{2},{4,7},{5,8}}
=> ? = 3
{{1,3},{2,5},{4,6},{7,8}}
=> {{1,8},{2,3,6},{4,5},{7}}
=> {{1,3,5},{2,6},{4,8},{7}}
=> {{1,3,5},{2},{4,7},{6,8}}
=> ? = 4
{{1,2},{3,5},{4,6},{7,8}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> {{1,2,5,6},{3,8},{4},{7}}
=> {{1,4,7,8},{2},{3,6},{5}}
=> ? = 5
{{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> {{1,7,8},{2},{3,5},{4,6}}
=> ? = 5
{{1,3},{2,6},{4,5},{7,8}}
=> {{1,5,8},{2,3},{4,6},{7}}
=> {{1,3},{2,8},{4,5,6},{7}}
=> {{1,6,8},{2},{3,4,5},{7}}
=> ? = 4
{{1,4},{2,6},{3,5},{7,8}}
=> {{1,8},{2,5},{3,4,6},{7}}
=> {{1,4,6},{2,5},{3,8},{7}}
=> {{1,3,7},{2},{4,6},{5,8}}
=> ? = 3
{{1,5},{2,6},{3,4},{7,8}}
=> {{1,4,8},{2},{3,5,6},{7}}
=> {{1,6},{2},{3,4,5,8},{7}}
=> {{1,3,8},{2},{4,5,6},{7}}
=> ? = 3
{{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> {{1,8},{2,6},{3,4,5},{7}}
=> {{1,8},{2},{3,7},{4,5,6}}
=> ? = 3
{{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5,8},{3},{6,7}}
=> {{1,5,7},{2,4},{3},{6,8}}
=> {{1,2,6},{3},{4,8},{5,7}}
=> ? = 2
{{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> {{1,8},{2,4,5},{3},{6,7}}
=> {{1,8},{2,3},{4,6},{5,7}}
=> ? = 2
{{1,8},{2,6},{3,4},{5,7}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> {{1,8},{2,6},{3,4},{5,7}}
=> {{1,8},{2,3,7},{4},{5,6}}
=> ? = 1
{{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2},{3,6,8},{5,7}}
=> {{1,8},{2},{3,4},{5,6,7}}
=> {{1,8},{2,3,4},{5,6},{7}}
=> ? = 1
{{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2},{3,8},{5,6,7}}
=> {{1,6,7},{2},{3,4},{5,8}}
=> {{1,2,7},{3,8},{4},{5,6}}
=> ? = 1
{{1,5},{2,7},{3,4},{6,8}}
=> {{1,4},{2,8},{3,5},{6,7}}
=> {{1,5},{2,7},{3,4},{6,8}}
=> {{1,2,5,6},{3},{4,8},{7}}
=> ? = 2
{{1,4},{2,7},{3,5},{6,8}}
=> {{1},{2,5,8},{3,4},{6,7}}
=> {{1},{2,4},{3,5,7},{6,8}}
=> {{1,2,4,5,7},{3},{6},{8}}
=> ? = 2
{{1,3},{2,7},{4,5},{6,8}}
=> {{1,5},{2,3,8},{4},{6,7}}
=> {{1,3,7},{2,5},{4},{6,8}}
=> {{1,2,7},{3},{4,6,8},{5}}
=> ? = 3
{{1,2},{3,7},{4,5},{6,8}}
=> {{1,2,5},{3,8},{4,7},{6}}
=> {{1,2,7},{3,8},{4,5},{6}}
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? = 4
{{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> {{1,2,7},{3},{4,5,8},{6}}
=> {{1,3},{2,7,8},{4,5},{6}}
=> ? = 4
{{1,3},{2,8},{4,5},{6,7}}
=> {{1,5,7},{2,3},{4},{6,8}}
=> {{1,3},{2,5,8},{4},{6,7}}
=> {{1,5},{2,3},{4,6,8},{7}}
=> ? = 3
{{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5},{3,8},{6,7}}
=> {{1,4,6},{2,3},{5,8},{7}}
=> ? = 2
{{1,5},{2,8},{3,4},{6,7}}
=> {{1,4,7},{2},{3,5},{6,8}}
=> {{1,8},{2},{3,4,5},{6,7}}
=> {{1,8},{2,3},{4,5,6},{7}}
=> ? = 2
{{1,6},{2,8},{3,4},{5,7}}
=> {{1,4},{2,7},{3},{5,6,8}}
=> {{1,6,8},{2,4},{3},{5,7}}
=> {{1,6},{2,3,8},{4},{5,7}}
=> ? = 1
{{1,7},{2,8},{3,4},{5,6}}
=> {{1,4,6},{2},{3},{5,7,8}}
=> {{1,4,7,8},{2},{3},{5,6}}
=> {{1,6},{2,7},{3,4},{5,8}}
=> ? = 1
{{1,8},{2,7},{3,4},{5,6}}
=> {{1,4,6},{2},{3,7},{5,8}}
=> {{1,8},{2},{3,4,6},{5,7}}
=> {{1,8},{2,3,5,6},{4},{7}}
=> ? = 1
{{1,8},{2,7},{3,5},{4,6}}
=> {{1},{2,5,6},{3,7},{4,8}}
=> {{1},{2,6},{3,8},{4,5,7}}
=> {{1,4,5},{2,3,7},{6},{8}}
=> ? = 0
{{1,7},{2,8},{3,5},{4,6}}
=> {{1},{2,5,6},{3},{4,7,8}}
=> {{1},{2,6},{3},{4,5,7,8}}
=> {{1,2,3,7},{4,5},{6},{8}}
=> ? = 0
{{1,6},{2,8},{3,5},{4,7}}
=> {{1},{2,5},{3,7},{4,6,8}}
=> {{1},{2,6,8},{3,7},{4,5}}
=> {{1,6},{2,4,5},{3,7},{8}}
=> ? = 0
{{1,5},{2,8},{3,6},{4,7}}
=> {{1},{2},{3,6,7},{4,5,8}}
=> {{1},{2},{3,5,7},{4,6,8}}
=> {{1,2,5},{3,4,6},{7},{8}}
=> ? = 0
{{1,4},{2,8},{3,6},{5,7}}
=> {{1},{2,7},{3,4,6},{5,8}}
=> {{1},{2,4,8},{3,6},{5,7}}
=> {{1,6},{2,3,5,7},{4},{8}}
=> ? = 1
{{1,3},{2,8},{4,6},{5,7}}
=> {{1},{2,3,6,7},{4},{5,8}}
=> {{1},{2,3,7},{4},{5,6,8}}
=> {{1,2,6,7},{3,4},{5},{8}}
=> ? = 2
{{1,2},{3,8},{4,6},{5,7}}
=> {{1,2},{3,6,7},{4,8},{5}}
=> {{1,2},{3,7},{4,6,8},{5}}
=> {{1,3,5},{2,6},{4},{7,8}}
=> ? = 3
{{1,2},{3,7},{4,6},{5,8}}
=> {{1,2},{3,6},{4,7,8},{5}}
=> {{1,2},{3,7,8},{4,6},{5}}
=> {{1,4},{2,6},{3,5},{7,8}}
=> ? = 3
{{1,3},{2,7},{4,6},{5,8}}
=> {{1},{2,3,6},{4,8},{5,7}}
=> {{1},{2,3,7},{4,8},{5,6}}
=> {{1,3,4},{2,6,7},{5},{8}}
=> ? = 2
{{1,4},{2,7},{3,6},{5,8}}
=> {{1},{2},{3,4,6,8},{5,7}}
=> {{1},{2},{3,4,8},{5,6,7}}
=> {{1,5,6},{2,3,4},{7},{8}}
=> ? = 1
{{1,5},{2,7},{3,6},{4,8}}
=> {{1},{2},{3,6},{4,5,7,8}}
=> {{1},{2},{3,5,7,8},{4,6}}
=> {{1,3,4,6},{2,5},{7},{8}}
=> ? = 0
{{1,6},{2,7},{3,5},{4,8}}
=> {{1},{2,5},{3},{4,6,7,8}}
=> {{1},{2,6,7,8},{3},{4,5}}
=> {{1,4,5},{2,6},{3,7},{8}}
=> ? = 0
{{1,7},{2,6},{3,5},{4,8}}
=> {{1},{2,5},{3,6},{4,7,8}}
=> {{1},{2,7,8},{3,6},{4,5}}
=> {{1,4,5},{2,7},{3,6},{8}}
=> ? = 0
{{1,8},{2,6},{3,5},{4,7}}
=> {{1},{2,5},{3,6,7},{4,8}}
=> {{1},{2,7},{3,6,8},{4,5}}
=> {{1,3,6},{2,7},{4,5},{8}}
=> ? = 0
{{1,8},{2,5},{3,6},{4,7}}
=> {{1},{2},{3,5,6,7},{4,8}}
=> {{1},{2},{3,6,7},{4,5,8}}
=> {{1,2,4,5},{3,6},{7},{8}}
=> ? = 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St001868
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 16%
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 16%
Values
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,5,2,1,3] => ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => [4,5,3,1,2] => ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => [4,2,5,1,3] => ? = 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 4
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 3
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => [4,1,2,3,6,5] => ? = 3
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 6
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [6,1,2,4,3,5] => [6,1,2,4,3,5] => ? = 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,1,2,6,3,4] => [5,1,2,6,3,4] => ? = 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => ? = 5
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,1,2,5,4,3] => [6,1,2,5,4,3] => ? = 2
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => [3,1,2,6,4,5] => ? = 2
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => [3,1,2,5,4,6] => ? = 5
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 4
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,1,2,6,5,4] => [3,1,2,6,5,4] => ? = 4
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Matching statistic: St001438
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 26%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 26%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[2],[]]
=> 0
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[1,1],[]]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> 0
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? = 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? = 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? = 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? = 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? = 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? = 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? = 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> ? = 4
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> ? = 3
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ? = 3
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> ? = 6
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ? = 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? = 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> ? = 5
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 2
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ? = 2
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ? = 5
Description
The number of missing boxes of a skew partition.
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