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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St000459
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => [2]
=> 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,1,1]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,1]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,1,1]
=> 4
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [2,1,1,1]
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [2,2,1]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,2,1]
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,2,1]
=> 4
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,2,1]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 5
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [2,1,1,1,1]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [2,2,1,1]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [2,1,1,1,1]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [2,2,1,1]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [2,2,1,1]
=> 5
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,6,3] => [2,2,1,1]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => [2,1,1,1,1]
=> 6
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [2,2,1,1]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,1,6,3,4] => [2,2,1,1]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [2,1,1,1,1]
=> 6
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [2,1,1,1,1]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [2,2,1,1]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [2,2,1,1]
=> 5
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [2,2,2]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [2,2,1,1]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => [2,2,1,1]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => [2,2,1,1]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => [2,2,1,1]
=> 5
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 6
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => [2,2,1,1]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => [2,1,1,1,1]
=> 6
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000691
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 98%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => => => ? = 2 - 3
[1,0,1,0]
=> [2,1] => 1 => 1 => 0 = 3 - 3
[1,1,0,0]
=> [1,2] => 0 => 0 => 0 = 3 - 3
[1,0,1,0,1,0]
=> [2,3,1] => 10 => 01 => 1 = 4 - 3
[1,0,1,1,0,0]
=> [2,1,3] => 10 => 01 => 1 = 4 - 3
[1,1,0,0,1,0]
=> [1,3,2] => 01 => 00 => 0 = 3 - 3
[1,1,0,1,0,0]
=> [3,1,2] => 10 => 01 => 1 = 4 - 3
[1,1,1,0,0,0]
=> [1,2,3] => 00 => 10 => 1 = 4 - 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 100 => 101 => 2 = 5 - 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 100 => 101 => 2 = 5 - 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 101 => 001 => 1 = 4 - 3
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 100 => 101 => 2 = 5 - 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 100 => 101 => 2 = 5 - 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 010 => 100 => 1 = 4 - 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 010 => 100 => 1 = 4 - 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 110 => 011 => 1 = 4 - 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 100 => 101 => 2 = 5 - 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 100 => 101 => 2 = 5 - 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 001 => 110 => 1 = 4 - 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 100 => 1 = 4 - 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 100 => 101 => 2 = 5 - 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 010 => 2 = 5 - 3
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1001 => 1101 => 2 = 5 - 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1010 => 1001 => 2 = 5 - 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1010 => 1001 => 2 = 5 - 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1010 => 1001 => 2 = 5 - 3
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 1101 => 2 = 5 - 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1010 => 1001 => 2 = 5 - 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1000 => 0101 => 3 = 6 - 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1000 => 0101 => 3 = 6 - 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0100 => 0100 => 2 = 5 - 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0100 => 0100 => 2 = 5 - 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0101 => 1100 => 1 = 4 - 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 0100 => 0100 => 2 = 5 - 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0100 => 0100 => 2 = 5 - 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 1100 => 1011 => 2 = 5 - 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1100 => 1011 => 2 = 5 - 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 1100 => 1011 => 2 = 5 - 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1000 => 0101 => 3 = 6 - 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 1000 => 0101 => 3 = 6 - 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 1101 => 2 = 5 - 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1100 => 1011 => 2 = 5 - 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 1000 => 0101 => 3 = 6 - 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1000 => 0101 => 3 = 6 - 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0010 => 0110 => 2 = 5 - 3
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,4,5,3,6,7,8] => ? => ? => ? = 8 - 3
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => 1000000000 => 0101010101 => ? = 12 - 3
[1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,3,8,9,2,4,5,6,7] => ? => ? => ? = 9 - 3
[]
=> [] => => => ? = 1 - 3
[1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,5,6,9,1,2,3,7,8] => ? => ? => ? = 10 - 3
[1,1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,5,8,9,1,2,3,6,7] => ? => ? => ? = 10 - 3
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,7,8,9,1,2,3,5,6] => ? => ? => ? = 10 - 3
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,8,9,1,2,3,4,7] => ? => ? => ? = 10 - 3
[1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0]
=> [6,7,1,8,9,2,3,4,5] => ? => ? => ? = 9 - 3
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> [6,7,8,1,2,9,3,4,5] => ? => ? => ? = 9 - 3
[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> [6,7,1,2,3,4,5,8,9] => ? => ? => ? = 10 - 3
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> [7,8,1,2,3,4,9,5,6] => ? => ? => ? = 9 - 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => 1000000000 => 0101010101 => ? = 12 - 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,10,11,1,2] => 1000000000 => 0101010101 => ? = 12 - 3
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,8,9,10,1,2,3,4,5,11] => 1000000000 => 0101010101 => ? = 12 - 3
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,7,8,9,10,11,2,3,4,5,6] => 0100000000 => 0101010100 => ? = 11 - 3
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => 0100000000 => 0101010100 => ? = 11 - 3
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => 1000000000 => 0101010101 => ? = 12 - 3
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000393
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 95%●distinct values known / distinct values provided: 75%
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 95%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => => ? = 2 - 2
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1 = 3 - 2
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 1 = 3 - 2
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 11 => 2 = 4 - 2
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 10 => 2 = 4 - 2
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 01 => 1 = 3 - 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 2 = 4 - 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 111 => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 110 => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => 101 => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 110 => 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 100 => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 011 => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 010 => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 101 => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 110 => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,1,2,4] => 100 => 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 001 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 010 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,4,3,1,2] => 1110 => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,2,3,1] => 1101 => 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,2,1,3] => 1110 => 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 1100 => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,3,4,2,1] => 1011 => 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,3,4,1,2] => 1010 => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => 1101 => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 1110 => 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1100 => 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => 1001 => 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => 1010 => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 1100 => 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => 0111 => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,5,3,1,2] => 0110 => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,5,2,3,1] => 0101 => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,5,2,1,3] => 0110 => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => 0100 => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => 1011 => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,5,1,2] => 1010 => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 1101 => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1110 => 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,1,2,5] => 1100 => 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,2,3,5,1] => 1001 => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => 1010 => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => 1100 => 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 0011 => 3 = 5 - 2
[1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,7,1,8] => ? => ? => ? = 9 - 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [7,5,4,3,1,2,6,8] => ? => ? => ? = 7 - 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => 1000000000 => ? = 12 - 2
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,3,2,5,6,7,8,9,1] => [9,3,2,1,4,5,6,7,8] => ? => ? = 10 - 2
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [5,4,2,3,6,7,8,9,1] => [9,3,4,2,1,5,6,7,8] => ? => ? = 9 - 2
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,6,7,8,9,1] => [9,4,3,2,1,5,6,7,8] => ? => ? = 10 - 2
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,6,7,8,9,10,1] => [10,4,3,2,1,5,6,7,8,9] => ? => ? = 11 - 2
[1,1,0,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,6,7,8,9,1,2] => ? => ? => ? = 9 - 2
[]
=> [] => [] => ? => ? = 1 - 2
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [6,5,7,8,4,3,2,1,9] => ? => ? => ? = 10 - 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [6,5,4,7,8,3,2,1,9] => ? => ? => ? = 10 - 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [6,5,4,3,7,8,2,1,9] => ? => ? => ? = 10 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [6,5,4,3,2,7,8,1,9] => ? => ? => ? = 10 - 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,7,5,4,3,2,1,8,9] => [7,6,5,4,3,1,2,8,9] => ? => ? = 10 - 2
[1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [5,4,6,7,3,2,1,8,9] => ? => ? => ? = 10 - 2
[1,1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,6,7,2,1,8,9] => ? => ? => ? = 10 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,4,3,2,7,1,8,9] => ? => ? => ? = 10 - 2
[1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,6,7,1,8,9] => ? => ? => ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [5,4,6,3,2,1,7,8,9] => ? => ? => ? = 10 - 2
[1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,6,2,1,7,8,9] => [6,5,3,2,1,4,7,8,9] => ? => ? = 10 - 2
[1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,4,3,2,6,1,7,8,9] => [6,4,3,2,1,5,7,8,9] => ? => ? = 10 - 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0]
=> [7,5,4,3,1,2,6,8,9] => ? => ? => ? = 8 - 2
[1,1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0,0]
=> [6,5,4,2,1,3,7,8,9] => ? => ? => ? = 9 - 2
[1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0]
=> [6,5,3,2,1,4,7,8,9] => [5,4,3,6,2,1,7,8,9] => ? => ? = 9 - 2
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0,0]
=> [7,4,3,2,1,5,6,8,9] => ? => ? => ? = 9 - 2
[1,1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0]
=> [3,4,5,2,1,6,7,8,9] => ? => ? => ? = 10 - 2
[1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [5,4,2,1,3,6,7,8,9] => [4,3,5,2,1,6,7,8,9] => ? => ? = 9 - 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> [8,3,2,1,4,5,6,7,9] => [4,3,2,5,6,7,8,1,9] => ? => ? = 9 - 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [10,11,9,8,7,6,5,4,3,2,1] => [11,10,9,8,7,6,5,4,3,1,2] => 1111111110 => ? = 12 - 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => 1111111110 => ? = 12 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,6,5,8,9,4,3,2,1] => ? => ? => ? = 10 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,2,1] => [9,8,6,7,5,4,3,2,1] => ? => ? = 9 - 2
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5,6,7,8,9,3,1,2] => [8,9,7,1,2,3,4,5,6] => ? => ? = 9 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [10,9,8,6,7,5,4,3,2,1] => [10,9,8,7,6,4,5,3,2,1] => ? => ? = 10 - 2
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1,8,9,10,11] => ? => ? => ? = 12 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [6,7,5,4,3,2,1,8,9,10] => ? => ? => ? = 11 - 2
[1,1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [7,6,5,4,3,1,2,8,9,10,11] => ? => ? => ? = 11 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,10,9,8,7,6,5,4,3,1,2] => [10,11,9,8,7,6,5,4,3,2,1] => 0111111111 => ? = 11 - 2
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => 1000000000 => ? = 12 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,3,2,5,1] => [9,7,6,5,8,4,3,2,1] => ? => ? = 9 - 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,3,5,2,1] => [9,8,6,5,7,4,3,2,1] => ? => ? = 9 - 2
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St001622
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([],1)
=> 0 = 2 - 2
[1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ?
=> ? = 7 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ?
=> ? = 7 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ?
=> ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 7 - 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 7 - 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,.],[[.,[.,[.,[.,.]]]],.]],.]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 7 - 2
[1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [[[.,[.,[.,[.,.]]]],[[.,.],.]],.]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [[[[[.,.],.],.],[.,.]],[.,[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [[[[[.,[.,.]],[[.,.],.]],.],.],.]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(6,4),(7,3)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ?
=> ? = 7 - 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001615
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([],1)
=> 0 = 2 - 2
[1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[.,[.,.]],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [[[.,[.,.]],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [[[.,[.,.]],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ?
=> ? = 7 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ?
=> ? = 7 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ?
=> ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 7 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 8 - 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 7 - 2
Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 67% ●values known / values provided: 73%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [.,.]
=> ([],1)
=> ([],1)
=> 0 = 2 - 2
[1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 3 - 2
[1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 6 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[[.,[.,.]],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [[[.,[.,.]],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [[[.,[.,.]],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 7 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ?
=> ? = 7 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ?
=> ? = 6 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
=> ?
=> ? = 7 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
=> ?
=> ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 7 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 8 - 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ([(0,5),(1,6),(2,7),(3,7),(4,3),(5,4),(6,2)],8)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ?
=> ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> ([(0,6),(1,3),(2,7),(3,7),(4,5),(5,2),(6,4)],8)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 8 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ([(0,6),(1,3),(2,4),(3,7),(4,5),(5,6),(6,7)],8)
=> ?
=> ? = 7 - 2
Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$
v(a\vee b) + v(a\wedge b) = v(a) + v(b).
$$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]].
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St000308
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 72%●distinct values known / distinct values provided: 58%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 72%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [[1]]
=> [1] => [] => ? = 2 - 2
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [1] => 1 = 3 - 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1] => 1 = 3 - 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2] => 2 = 4 - 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [2,1] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2 = 4 - 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,2] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3] => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,3] => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,1,2] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,2,3] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [1,4,2,3] => 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [1,4,2,3] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => [3,1,2,4] => 3 = 5 - 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? => ? => ? = 7 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => ? => ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6] => ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,5,6,8] => ? => ? = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 9 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ? => ? = 8 - 2
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 9 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,7,3,4,5,6,8] => ? => ? = 8 - 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? => ? = 9 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,3,2,4,5,6,8,7] => ? => ? = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 7 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 9 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,4,2,3,5,6,7,8] => [1,4,2,3,5,6,7] => ? = 8 - 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ? => ? = 8 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 8 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 6 - 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,-1,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? => ? = 9 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7,8] => [1,6,2,3,4,5,7] => ? = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,6,2,3,4,5,8,7] => ? => ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,4,2,3,8,5,6,7] => [1,4,2,3,5,6,7] => ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,-1,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,3,2,8,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 8 - 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,-1,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? => ? = 9 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [2,1,3,4,5,7,6,8] => ? => ? = 7 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,8,6,7] => ? => ? = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [2,1,7,3,4,5,6,8] => ? => ? = 7 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,3,2,4,5,6,7,8] => [1,3,2,4,5,6,7] => ? = 8 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,3,2,4,5,6,8,7] => ? => ? = 8 - 2
[1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,3,2,4,8,5,6,7] => [1,3,2,4,5,6,7] => ? = 8 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7,8] => ? => ? = 8 - 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7] => ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => ? => ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6] => ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ? => ? = 8 - 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,5,6,8] => ? => ? = 8 - 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [3,1,2,4,5,6,8,7] => ? => ? = 8 - 2
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 8 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 7 - 2
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,8,6,7] => ? => ? = 8 - 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,4,2,3,5,6,7,8] => [1,4,2,3,5,6,7] => ? = 8 - 2
[1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,5,3,4,6,7,8] => ? => ? = 8 - 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000702
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 71%●distinct values known / distinct values provided: 58%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 71%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [[1]]
=> [1] => [] => ? = 2 - 2
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => [1] => ? = 3 - 2
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => [1] => ? = 3 - 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [1,2] => 2 = 4 - 2
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2 = 4 - 2
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [2,1] => 1 = 3 - 2
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [1,2] => 2 = 4 - 2
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [1,2] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3] => 3 = 5 - 2
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 5 - 2
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 5 - 2
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [2,1,3] => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [1,3,2] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => 3 = 5 - 2
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 5 - 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,1,2] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [2,1,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => 3 = 5 - 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,2,3] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 5 - 2
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => 3 = 5 - 2
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [1,4,2,3] => 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => [2,1,3,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => [1,2,4,3] => 3 = 5 - 2
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => [1,4,2,3] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => [1,2,3,4] => 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => [3,1,2,4] => 3 = 5 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => [3,1,2,4] => 3 = 5 - 2
[1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? => ? => ? = 7 - 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => ? => ? = 8 - 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,5,6,8] => ? => ? = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 9 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ? => ? = 8 - 2
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 9 - 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,7,3,4,5,6,8] => ? => ? = 8 - 2
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? => ? = 9 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,3,2,4,5,6,8,7] => ? => ? = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 7 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 8 - 2
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 9 - 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ? => ? = 8 - 2
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 8 - 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 6 - 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,1,0,0,-1,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? => ? = 9 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,6,2,3,4,5,7,8] => [1,6,2,3,4,5,7] => ? = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,6,2,3,4,5,8,7] => ? => ? = 8 - 2
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 8 - 2
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,-1,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,8,3,4,5,6,7] => ? => ? = 9 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => ? = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => ? = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [2,1,3,4,5,7,6,8] => ? => ? = 7 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,8,6,7] => ? => ? = 8 - 2
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,8,5,6,7] => [2,1,3,4,5,6,7] => ? = 8 - 2
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,3,8,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 6 - 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => ? = 8 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [2,1,7,3,4,5,6,8] => ? => ? = 7 - 2
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [2,1,8,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 8 - 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,3,2,4,5,6,8,7] => ? => ? = 8 - 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7,8] => ? => ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => ? => ? = 8 - 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ? => ? = 8 - 2
[1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,5,6,8] => ? => ? = 8 - 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7] => ? = 8 - 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [3,1,2,4,5,6,8,7] => ? => ? = 8 - 2
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ? => ? = 8 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? => ? => ? = 7 - 2
[1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [3,1,2,4,8,5,6,7] => [3,1,2,4,5,6,7] => ? = 8 - 2
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => ? = 8 - 2
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000093
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 66%●distinct values known / distinct values provided: 58%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 66%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => ([],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3] => ([],3)
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 5 = 6 - 1
[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] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [8] => ([],8)
=> ? = 9 - 1
[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] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,6,1,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6,8] => ? => ?
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,5,1,6,8,7] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,7,8,1,5,6] => ? => ?
=> ? = 9 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,5,6,7,4,8] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,1,4] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,6,7,8,1,4,5] => ? => ?
=> ? = 9 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,1,4,5,6,8,7] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => ? => ?
=> ? = 9 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,7,3,8] => ? => ?
=> ? = 8 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,1,3] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,6,7,8,1,3,5] => ? => ?
=> ? = 9 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,8,4] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,7,1,8,3,4] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,8,1,3,4] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,1,3,4,6,5,7,8] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,3,4,5,7,8,6] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,1,3,4,5,7,6,8] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => [1,7] => ([(6,7)],8)
=> ? = 8 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => ? => ?
=> ? = 8 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,6,2,8,7] => ? => ?
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,6,2,7,8] => ? => ?
=> ? = 8 - 1
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,4,8,2,5,6,7] => ? => ?
=> ? = 8 - 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,4,2,5,6,7,8] => ? => ?
=> ? = 8 - 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000786
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 66%●distinct values known / distinct values provided: 58%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 66%●distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2] => ([],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [2] => ([],2)
=> 2 = 3 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3] => ([],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [3] => ([],3)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3] => ([],3)
=> 3 = 4 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3] => ([],3)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => ([(2,3)],4)
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => ([],4)
=> 4 = 5 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => ([],4)
=> 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => ([],5)
=> 5 = 6 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5] => ([],5)
=> 5 = 6 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5] => ([],5)
=> 5 = 6 - 1
[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] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [8] => ([],8)
=> ? = 9 - 1
[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] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,6,1,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,5,7,1,8,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,5,7,1,6,8] => ? => ?
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,5,1,6,8,7] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,5,8,1,6,7] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1,6,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,4,7,8,1,5,6] => ? => ?
=> ? = 9 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,5,6,7,4,8] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,1,4] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,6,7,8,1,4,5] => ? => ?
=> ? = 9 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,1,4,5,6,8,7] => ? => ?
=> ? = 8 - 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7,8] => ? => ?
=> ? = 9 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,7,3,8] => ? => ?
=> ? = 8 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 7 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,4,1,5,6,7,8,3] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,1,3] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,6,7,8,1,3,5] => ? => ?
=> ? = 9 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,8,4] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,7,1,8,3,4] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,8,1,3,4] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,1,3,4,6,5,7,8] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [2,1,6,3,7,4,8,5] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,1,3,4,5,7,8,6] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,1,3,4,5,7,6,8] => ? => ?
=> ? = 8 - 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,7,8,1,3,4,5,6] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,3,4,5,6,8,7] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,1,8,3,4,5,6,7] => [2,6] => ([(5,7),(6,7)],8)
=> ? = 8 - 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [8] => ([],8)
=> ? = 9 - 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [8] => ([],8)
=> ? = 9 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,8,2] => [1,7] => ([(6,7)],8)
=> ? = 8 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => ? => ?
=> ? = 8 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,6,2,8,7] => ? => ?
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,6,2,7,8] => ? => ?
=> ? = 8 - 1
[1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,3,4,8,2,5,6,7] => ? => ?
=> ? = 8 - 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,4,2,5,6,7,8] => ? => ?
=> ? = 8 - 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000062The length of the longest increasing subsequence of the permutation. St000991The number of right-to-left minima of a permutation. St001875The number of simple modules with projective dimension at most 1. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000619The number of cyclic descents of a permutation. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000863The length of the first row of the shifted shape of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001820The size of the image of the pop stack sorting operator. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001626The number of maximal proper sublattices of a lattice. St000907The number of maximal antichains of minimal length in a poset. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001649The length of a longest trail in a graph. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St001668The number of points of the poset minus the width of the poset. St000144The pyramid weight of the Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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