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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [2,1]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => [3]
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => [3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [3,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [3,2]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [4,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [3,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [3,2]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => [3,2]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => [4,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [3,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000392
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 89%●distinct values known / distinct values provided: 85%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 89%●distinct values known / distinct values provided: 85%
Values
[1,0]
=> [1] => [1] => => ? = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 01 => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 11 => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => 011 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 011 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 110 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 011 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0001 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 0101 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 0011 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1001 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 0101 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 0011 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 0110 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 0111 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0001 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0010 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 0011 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 1001 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 1010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => 0101 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 0011 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 0110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 1101 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 1011 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 0111 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 1100 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0001 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7,8] => [6,5,1,2,3,4,7,8] => ? => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,5,6,4,7,8] => [3,1,2,6,4,5,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,7,8,4,6] => [3,1,2,7,4,5,8,6] => ? => ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,5,4,6,7,8] => [3,1,2,5,4,6,7,8] => ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,5,1,6,4,8,7] => [6,4,1,2,3,5,8,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,7,1,8,4] => [8,4,6,1,2,3,5,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,3,5,1,7,4,8,6] => [8,6,4,1,2,3,5,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,1,6,4,7,8,5] => [3,1,2,8,5,4,6,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,3,1,6,7,4,8,5] => [3,1,2,6,4,8,5,7] => ? => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,3,6,7,1,4,8,5] => [8,5,1,2,3,6,4,7] => ? => ? = 3 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,1,4,5,8,6,7] => [3,1,2,4,5,8,7,6] => ? => ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,7,3,8,6] => [2,1,8,6,3,4,5,7] => ? => ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,4,7,3,8,5,6] => [2,1,7,5,3,4,8,6] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => [8,7,3,1,2,4,5,6] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,7,3,8,6] => [8,6,3,1,2,4,5,7] => ? => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,7,1,3,8,6] => [8,6,3,5,1,2,4,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,8,3,6,7] => [4,1,2,8,7,6,3,5] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,7,5,8] => [4,3,1,2,7,5,6,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,1,6,3,5,7,8] => [6,5,3,1,2,4,7,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,4,6,1,3,7,5,8] => [4,1,2,7,5,3,6,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,4,6,7,1,3,8,5] => [6,3,8,5,1,2,4,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,6,7,1,8,3,5] => [8,5,1,2,4,7,3,6] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5,8] => [6,3,7,5,1,2,4,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,4,1,3,5,7,8,6] => [4,3,1,2,5,8,6,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [4,3,1,2,7,5,8,6] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,4,1,3,7,5,6,8] => [4,3,1,2,7,6,5,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,4,1,7,3,5,8,6] => [8,6,5,3,1,2,4,7] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [2,4,7,1,3,5,8,6] => [4,1,2,8,6,5,3,7] => ? => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,7,8,6] => [2,1,3,5,4,8,6,7] => ? => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,7,8,4] => [2,1,8,4,3,5,6,7] => ? => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [2,1,5,6,3,4,7,8] => [2,1,5,3,6,4,7,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,7,4,6,8] => [2,1,7,6,4,3,5,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [2,1,5,3,4,6,8,7] => ? => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [2,5,1,3,6,7,8,4] => [8,4,3,1,2,5,6,7] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [2,5,1,3,6,8,4,7] => [8,7,4,3,1,2,5,6] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [2,5,1,6,3,4,8,7] => [5,3,1,2,6,4,8,7] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [2,5,6,1,3,7,4,8] => [7,4,1,2,5,3,6,8] => ? => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [2,5,6,1,7,3,8,4] => [6,3,8,4,1,2,5,7] => ? => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,5,1,3,7,8,4,6] => [7,4,3,1,2,5,8,6] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [2,1,3,6,7,4,8,5] => [2,1,3,6,4,8,5,7] => ? => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [2,1,6,7,3,4,8,5] => [2,1,8,5,3,6,4,7] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [2,6,1,3,7,8,4,5] => [8,5,7,4,3,1,2,6] => ? => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,6,1,7,3,4,8,5] => [8,5,3,1,2,6,4,7] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,6,2,7,8,5] => [1,8,5,2,3,4,6,7] => ? => ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,4,2,7,5,6,8] => [1,4,2,3,7,6,5,8] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,4,7,2,5,8,6] => [1,8,6,5,2,3,4,7] => ? => ? = 3 - 1
[1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6,8] => [1,7,6,5,2,3,4,8] => ? => ? = 4 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ? => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,5,6,7,2,8,4] => [1,8,4,6,2,3,5,7] => ? => ? = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000444
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 54%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 54%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,5,6,7,8,2] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,4,1,5,6,7,8,2] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,5,6,7,8,4] => [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,2,5,6,7,8,3] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,6,7,8,5] => [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,4,5,1,6,7,8,3] => ?
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,4,5,1,6,7,8,2] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,1,3,6,7,8,5] => [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [3,4,1,2,6,7,8,5] => [1,1,1,0,1,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,5,1,2,6,7,8,4] => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,6,7,8,5] => [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,6,7,8,5] => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [5,1,2,3,6,7,8,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [3,4,1,5,2,7,8,6] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,4,6,1,7,8,5] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,6,2,7,8,5] => ?
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,5,6,1,7,8,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,4,5,6,1,7,8,3] => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,6,1,7,8,2] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,5,6,2,7,8,3] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,1,6,4,7,8,5] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,1,6,3,7,8,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,5,1,6,2,7,8,3] => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [3,1,2,6,4,7,8,5] => [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,6,3,7,8,5] => [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,4,1,5,7,8,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,7,8,6] => [1,1,1,0,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,1,4,7,8,6] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,5,1,3,7,8,6] => [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [3,4,5,1,2,7,8,6] => [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,1,5,2,4,7,8,6] => [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,4,6,1,3,7,8,5] => [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [3,4,6,1,2,7,8,5] => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [2,5,6,1,3,7,8,4] => [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,4,1,2,5,7,8,6] => [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [2,5,1,3,4,7,8,6] => ?
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,5,1,2,4,7,8,6] => [1,1,1,0,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [5,6,1,2,3,7,8,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [3,1,2,4,5,7,8,6] => [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,7,8,6] => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,7,8,6] => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,7,8,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 6
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001062
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 62%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => {{1,2,3}}
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => {{1,3,4},{2}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => {{1,2,4},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => {{1,4},{2},{3,5}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => {{1,3,4},{2},{5}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => {{1,4},{2,5},{3}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [7,2,3,4,5,6,1,8] => {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,1,8,7] => [6,2,3,4,5,1,8,7] => {{1,6},{2},{3},{4},{5},{7,8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [7,2,3,4,5,6,8,1] => {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,6,1,7,8] => [6,2,3,4,5,1,7,8] => {{1,6},{2},{3},{4},{5},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => [5,2,3,4,1,8,7,6] => {{1,5},{2},{3},{4},{6,8},{7}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => [7,2,3,4,5,8,1,6] => {{1,7},{2},{3},{4},{5},{6,8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [6,2,3,4,5,8,7,1] => {{1,6,8},{2},{3},{4},{5},{7}}
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6,8] => [6,2,3,4,5,7,1,8] => {{1,6,7},{2},{3},{4},{5},{8}}
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,5,1,6,8,7] => [5,2,3,4,1,6,8,7] => {{1,5},{2},{3},{4},{6},{7,8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,5,1,8,6,7] => [5,2,3,4,1,7,8,6] => {{1,5},{2},{3},{4},{6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [6,2,3,4,5,7,8,1] => {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 4
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [5,2,3,4,1,6,7,8] => {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [4,2,3,1,8,6,7,5] => {{1,4},{2},{3},{5,8},{6},{7}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,6,7,5,8] => [4,2,3,1,7,6,5,8] => {{1,4},{2},{3},{5,7},{6},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,8,7] => [4,2,3,1,6,5,8,7] => {{1,4},{2},{3},{5,6},{7,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,4,1,6,8,5,7] => [4,2,3,1,7,6,8,5] => {{1,4},{2},{3},{5,7,8},{6}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,4,1,6,5,7,8] => [4,2,3,1,6,5,7,8] => {{1,4},{2},{3},{5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,4,6,1,7,8,5] => [6,2,3,4,8,1,7,5] => {{1,6},{2},{3},{4},{5,8},{7}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,6,7,1,8,5] => [7,2,3,4,8,6,1,5] => {{1,7},{2},{3},{4},{5,8},{6}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [5,2,3,4,8,6,7,1] => {{1,5,8},{2},{3},{4},{6},{7}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,7,1,5,8] => [5,2,3,4,7,6,1,8] => {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,4,6,1,5,8,7] => [5,2,3,4,6,1,8,7] => {{1,5,6},{2},{3},{4},{7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,6,1,5,7,8] => [5,2,3,4,6,1,7,8] => {{1,5,6},{2},{3},{4},{7},{8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,4,1,5,7,8,6] => [4,2,3,1,5,8,7,6] => {{1,4},{2},{3},{5},{6,8},{7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,4,1,7,5,8,6] => [4,2,3,1,7,8,5,6] => {{1,4},{2},{3},{5,7},{6,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,4,1,7,8,5,6] => [4,2,3,1,6,8,7,5] => {{1,4},{2},{3},{5,6,8},{7}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,4,7,1,8,5,6] => [7,2,3,4,6,8,1,5] => {{1,7},{2},{3},{4},{5,6,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,7,1,5,6,8] => [5,2,3,4,6,7,1,8] => {{1,5,6,7},{2},{3},{4},{8}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,4,1,5,8,6,7] => [4,2,3,1,5,7,8,6] => {{1,4},{2},{3},{5},{6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,4,1,8,5,6,7] => [4,2,3,1,6,7,8,5] => {{1,4},{2},{3},{5,6,7,8}}
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,4,8,1,5,6,7] => [5,2,3,4,6,7,8,1] => {{1,5,6,7,8},{2},{3},{4}}
=> ? = 5
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [4,2,3,1,5,6,7,8] => {{1,4},{2},{3},{5},{6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => [3,2,1,8,5,6,7,4] => {{1,3},{2},{4,8},{5},{6},{7}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,5,6,7,4,8] => [3,2,1,7,5,6,4,8] => {{1,3},{2},{4,7},{5},{6},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => [3,2,1,6,5,4,8,7] => {{1,3},{2},{4,6},{5},{7,8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,3,1,5,6,4,7,8] => [3,2,1,6,5,4,7,8] => {{1,3},{2},{4,6},{5},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6,8] => [3,2,1,5,4,7,6,8] => {{1,3},{2},{4,5},{6,7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,7,8,4,6] => [3,2,1,6,5,8,7,4] => {{1,3},{2},{4,6,8},{5},{7}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,3,1,5,7,4,6,8] => [3,2,1,6,5,7,4,8] => {{1,3},{2},{4,6,7},{5},{8}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,5,4,6,7,8] => [3,2,1,5,4,6,7,8] => {{1,3},{2},{4,5},{6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,5,1,6,7,8,4] => [5,2,3,8,1,6,7,4] => {{1,5},{2},{3},{4,8},{6},{7}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,5,1,6,7,4,8] => [5,2,3,7,1,6,4,8] => {{1,5},{2},{3},{4,7},{6},{8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,5,1,6,4,8,7] => [5,2,3,6,1,4,8,7] => {{1,5},{2},{3},{4,6},{7,8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,4,7,8] => [5,2,3,6,1,4,7,8] => {{1,5},{2},{3},{4,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,5,6,1,7,8,4] => [6,2,3,8,5,1,7,4] => {{1,6},{2},{3},{4,8},{5},{7}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,5,6,1,7,4,8] => [6,2,3,7,5,1,4,8] => {{1,6},{2},{3},{4,7},{5},{8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,7,1,8,4] => [7,2,3,8,5,6,1,4] => {{1,7},{2},{3},{4,8},{5},{6}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,1,4] => [4,2,3,8,5,6,7,1] => {{1,4,8},{2},{3},{5},{6},{7}}
=> ? = 3
Description
The maximal size of a block of a set partition.
Matching statistic: St000485
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 46%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1] => [1] => [1] => ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [2,3,1] => 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => [3,2,4,1] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,1,3] => [3,2,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,3,5,2,4] => [1,4,3,5,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,4,3,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3,5] => [3,2,4,1,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,3,4] => [3,2,4,5,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,3,5,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,1,3,5] => [3,2,4,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [2,3,1,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [3,4,1,5,2] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,5,1,2,4] => [2,4,3,5,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [1,3,5,2,7,4,6] => [1,5,3,6,2,7,4] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [1,3,5,2,7,4,6] => [1,5,3,6,2,7,4] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [2,1,3,5,7,4,6] => [2,1,3,6,5,7,4] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,6,7,3,5] => [2,1,3,5,7,4,6] => [2,1,3,6,5,7,4] => ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => [2,1,4,6,3,5,7] => [2,1,5,4,6,3,7] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 3
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,4,7,3,5,6] => [2,1,4,7,3,5,6] => [2,1,5,4,6,7,3] => ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => [1,3,5,2,7,4,6] => [1,5,3,6,2,7,4] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,1,3,6,7,5] => [2,4,1,3,5,7,6] => [3,2,4,1,5,7,6] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [2,4,1,3,6,5,7] => [3,2,4,1,6,5,7] => ? = 3
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => [2,4,1,6,3,5,7] => [4,2,5,1,6,3,7] => ? = 3
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,6,1,7,3,5] => [1,3,5,2,7,4,6] => [1,5,3,6,2,7,4] => ? = 3
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => [2,4,6,1,3,5,7] => [3,2,5,4,6,1,7] => ? = 4
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [2,4,1,3,5,7,6] => [3,2,4,1,5,7,6] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,4,1,3,7,5,6] => [2,4,1,3,7,5,6] => [3,2,4,1,6,7,5] => ? = 3
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,4,1,7,3,5,6] => [2,4,1,7,3,5,6] => [4,2,5,1,6,7,3] => ? = 4
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => [2,4,7,1,3,5,6] => [3,2,5,4,6,7,1] => ? = 5
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,4,1,3,5,6,7] => [2,4,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,1,3,5,7,4,6] => [2,1,3,5,7,4,6] => [2,1,3,6,5,7,4] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,6,7,3,4] => [2,1,3,5,7,4,6] => [2,1,3,6,5,7,4] => ? = 3
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,1,5,6,3,4,7] => [2,1,4,6,3,5,7] => [2,1,5,4,6,3,7] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,4,7,6] => [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,6] => [2,1,5,3,7,4,6] => [2,1,5,6,3,7,4] => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,1,5,7,3,4,6] => [2,1,5,7,3,4,6] => [2,1,4,6,5,7,3] => ? = 4
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,4,5,3,6,7] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,1,3,6,7,4] => [2,4,1,3,5,7,6] => [3,2,4,1,5,7,6] => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [2,4,1,3,6,5,7] => [3,2,4,1,6,5,7] => ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => [2,4,1,6,3,5,7] => [4,2,5,1,6,3,7] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => [1,3,5,2,7,4,6] => [1,5,3,6,2,7,4] => ? = 3
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [2,4,6,1,3,5,7] => [3,2,5,4,6,1,7] => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [2,5,1,3,4,7,6] => [3,2,4,5,1,7,6] => ? = 4
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,5,1,3,7,4,6] => [2,5,1,3,7,4,6] => [3,2,5,6,1,7,4] => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => [2,5,1,7,3,4,6] => [5,2,4,6,1,7,3] => ? = 4
Description
The length of the longest cycle of a permutation.
Matching statistic: St001239
St001239: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001192
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
St001192: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001235
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,1] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [5,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,3,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [3,2,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [4,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,1,1] => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [4,1,2] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,3,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,2,1,1] => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,1,1,1] => ? = 4
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,2] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,2,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,1,1] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [2,2,3] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [3,3,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [3,2,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,2,1] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,1,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [4,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,1,2,1] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [3,2,1,1] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,1,1,1] => ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,1,3] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,4,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,2] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,2,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [2,2,1,2] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,2,1,1] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,1,1,1] => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [3,1,1,2] => ? = 4
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,4,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,1,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,2,1,1,1] => ? = 4
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,1,1,1,1] => ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,5] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,4,2] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,3,2,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,4,1,1] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,3,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,2,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,3,2,1] => ? = 2
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001232
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 54%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 54%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000374
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 23%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 23%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,4,5,6] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,5,4,2,3,6] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [3,2,5,4,1,6,7,8] => ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,7,6,4,5,8] => ? = 3 - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [5,4,3,2,1,6,7,8] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,5,4,7,6,2,3,8] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,5,4,2,3,6,7,8] => ? = 3 - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,7,6,5,4,2,3,8] => ? = 3 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,7,6,3,4,5,8] => ? = 4 - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [3,2,5,4,7,6,9,8,1,10] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [3,2,5,4,7,6,1,8,9,10] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [3,2,5,4,9,8,7,6,1,10] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [3,2,5,4,1,9,8,6,7,10] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [3,2,5,4,1,6,7,8,9,10] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [3,2,7,6,5,4,9,8,1,10] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [3,2,7,6,5,4,1,8,9,10] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [3,2,9,6,5,8,7,4,1,10] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [3,2,1,7,6,9,8,4,5,10] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [3,2,1,7,6,4,5,8,9,10] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [3,2,9,8,7,6,5,4,1,10] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [3,2,1,9,8,7,6,4,5,10] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [3,2,1,4,9,8,5,6,7,10] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [5,4,3,2,7,6,9,8,1,10] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [5,4,3,2,7,6,1,8,9,10] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [5,4,3,2,9,8,7,6,1,10] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [5,4,3,2,1,9,8,6,7,10] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,6,7,8,9,10] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [7,4,3,6,5,2,9,8,1,10] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [7,4,3,6,5,2,1,8,9,10] => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [9,4,3,6,5,8,7,2,1,10] => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,5,4,7,6,9,8,2,3,10] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,5,4,7,6,2,3,8,9,10] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [9,4,3,8,7,6,5,2,1,10] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,5,4,9,8,7,6,2,3,10] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,5,4,2,9,8,3,6,7,10] => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,5,4,2,3,6,7,8,9,10] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [7,6,5,4,3,2,9,8,1,10] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [7,6,5,4,3,2,1,8,9,10] => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [9,6,5,4,3,8,7,2,1,10] => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,7,6,5,4,9,8,2,3,10] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,7,6,5,4,2,3,8,9,10] => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [9,8,5,4,7,6,3,2,1,10] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,9,6,5,8,7,4,2,3,10] => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,2,7,6,9,8,3,4,5,10] => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [1,2,7,6,3,4,5,8,9,10] => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [1,9,8,7,6,5,4,2,3,10] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [1,2,9,8,7,6,3,4,5,10] => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [1,2,3,9,8,4,5,6,7,10] => ? = 5 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [3,2,5,4,7,6,9,8,11,10,1,12] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => [3,2,5,4,7,6,9,8,1,10,11,12] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,10,9,8,7,12,11] => [3,2,5,4,7,6,11,10,9,8,1,12] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,12,9,8,11,10,7] => [3,2,5,4,7,6,1,11,10,8,9,12] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,12,11,10,9,8,7] => [3,2,5,4,7,6,1,8,9,10,11,12] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,8,7,6,5,10,9,12,11] => [3,2,5,4,9,8,7,6,11,10,1,12] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,8,7,6,5,12,11,10,9] => [3,2,5,4,9,8,7,6,1,10,11,12] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,10,7,6,9,8,5,12,11] => [3,2,5,4,11,8,7,10,9,6,1,12] => ? = 2 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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