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Your data matches 157 different statistics following compositions of up to 3 maps.
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Matching statistic: St000676
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000288
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 89%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> [1,0]
=> [1] => => ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1 => 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 01 => 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 101 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 011 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 001 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 011 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 010 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 101 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 001 => 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1110 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1101 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1100 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 0111 => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1011 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1010 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1001 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1001 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 0111 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0110 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1011 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0101 => 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0111 => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0110 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0101 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0011 => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0011 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0001 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0001 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0011 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0101 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0001 => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0111 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,1,5,4,8,7,6] => ? => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,5,4,6,8,7] => ? => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,2,1,5,4,7,8,6] => ? => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,2,1,4,6,5,8,7] => ? => ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,2,1,4,6,5,7,8] => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [3,2,1,4,6,7,5,8] => ? => ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,2,1,5,6,4,8,7] => ? => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [4,3,2,7,6,5,1,8] => ? => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [4,3,2,5,1,7,6,8] => ? => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [4,3,2,5,1,6,8,7] => ? => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [4,3,2,7,6,8,5,1] => ? => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [5,4,3,7,6,8,2,1] => ? => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,6,7,8,5,1] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,1,5,4,3,7,8,6] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,6,5,4,7,3,8] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,6,5,4,7,8,3] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,4,7,6,3,8] => ? => ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,4,6,3,7,8] => ? => ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,1,6,5,7,4,3,8] => ? => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ? => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,5,4,7,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,7,6,5,8,4] => ? => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,3,6,5,7,4,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,3,7,6,8,5,4] => ? => ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,3,6,5,7,8,4] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,4,6,8,7,5] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,3,5,8,7,6,4] => ? => ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,6,4,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,1,3,6,7,5,8,4] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,3,5,6,8,7,4] => ? => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,6,7,4,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,3,5,7,8,6,4] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,1,4,7,6,5,3,8] => ? => ? = 5 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3,7,8,6] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,5,7,3,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,1,4,6,5,7,8,3] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,5,7,6,4,3,8] => ? => ? = 5 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,1,5,6,4,7,3,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,6,3,8] => ? => ? = 4 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,6,3,7,8] => ? => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,4,5,7,6,8,3] => ? => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0,1,0]
=> [3,2,6,5,4,1,7,8] => ? => ? = 5 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000157
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 89%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> [.,.]
=> [1] => [[1]]
=> 0 = 1 - 1
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 1 = 2 - 1
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => [[1,3],[2]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [6,7,8,4,5,3,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [8,5,4,6,7,3,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [6,4,3,5,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [6,7,4,3,5,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [6,7,3,4,5,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [7,8,4,5,3,6,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [8,5,4,6,3,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[.,[[.,.],[.,.]]],[.,.]]]]
=> [8,6,4,5,3,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [7,8,4,3,5,6,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [8,5,6,3,4,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[.,[[.,[[.,.],.]],.]]],.]]
=> [5,6,4,7,3,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,[[[.,[.,.]],.],.]]],.]]
=> [5,4,6,7,3,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],[.,.]]]],.]]
=> [7,5,4,6,3,2,8,1] => ?
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> [5,3,4,6,7,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,[[.,.],[.,[[.,.],.]]]],.]]
=> [6,7,5,3,4,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> [5,6,7,3,4,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [6,7,4,3,5,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [6,7,3,4,5,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [7,4,3,5,6,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> [5,6,4,3,2,7,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [.,[[[.,[.,[[.,.],[.,.]]]],.],.]]
=> [6,4,5,3,2,7,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[[[.,[[.,[[.,.],.]],.]],.],.]]
=> [4,5,3,6,2,7,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,[[[.,.],[.,.]],.]],.],.]]
=> [5,3,4,6,2,7,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,[[.,.],[[.,.],.]]],.],.]]
=> [5,6,3,4,2,7,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[[.,[.,.]],[.,.]]],.],.]]
=> [6,4,3,5,2,7,8,1] => ?
=> ? = 5 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000010
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 7
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,1,2,3] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,1,2,3] => ?
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,1,2,3] => ?
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,1,2,3] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,1,2,3] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,8,1,2,3] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,1,2,3,5] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,1,2,3,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,1,2,3,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,1,2,3,5,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,1,2,3,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [7,6,4,1,2,3,5,8] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,1,2,3,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,1,2,3,7] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,1,2,3,8] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,1,2,3,8] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,1,2,3,8] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,1,2,3,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,1,2,3,7,8] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,1,2,3,6,8] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,1,2] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,4,5,1,2] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,1,2] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,1,2] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,1,2] => ?
=> ? = 6
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,1,2] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [8,7,5,3,4,6,1,2] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,1,2] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,1,2] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,8,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,3,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,1,2] => ?
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,1,2] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,3,1,2] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,3,1,2] => ?
=> ? = 4
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,1,2] => ?
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,1,2] => ?
=> ? = 5
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ?
=> ? = 5
Description
The length of the partition.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 85%●distinct values known / distinct values provided: 78%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 85%●distinct values known / distinct values provided: 78%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,2] => ([(1,6),(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)
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [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)
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
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
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 85%●distinct values known / distinct values provided: 78%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 85%●distinct values known / distinct values provided: 78%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 7
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [4,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,2] => ([(1,6),(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)
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [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)
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5
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 $3$.
Matching statistic: St001176
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> [1,0]
=> [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,1]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [2]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [6,5,1,2,3,4,7,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,1,2,3] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,1,2,3] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,1,2,3] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,1,2,3] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,1,2,3] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [7,5,6,4,8,1,2,3] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,1,2,3,5] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,1,2,3,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,1,2,3,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,1,2,3,5,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,1,2,3,7,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [7,6,4,1,2,3,5,8] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,1,2,3,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,1,2,3,7] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,1,2,3,8] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,1,2,3,8] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,1,2,3,8] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,6,4,1,2,3,7,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,1,2,3,7,8] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [7,5,4,1,2,3,6,8] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,4,5,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,1,2] => ?
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,1,2] => ?
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [8,7,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,8,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,3,1,2] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,7,3,1,2] => ?
=> ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,1,2] => ?
=> ? = 5 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ?
=> ? = 5 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000147
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> [.,.]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [2]
=> 2
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ?
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [5,3,4,6,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [6,4,3,5,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [6,7,4,3,5,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [6,7,3,4,5,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,3,4,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [7,8,4,5,3,6,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [8,5,4,6,3,7,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [7,8,4,3,5,6,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [8,5,6,3,4,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [8,7,5,3,4,6,2,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [5,6,7,4,3,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[.,[[.,[[.,.],.]],.]]],.]]
=> [5,6,4,7,3,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,[[[.,[.,.]],.],.]]],.]]
=> [5,4,6,7,3,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],[.,.]]]],.]]
=> [7,5,4,6,3,2,8,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> [4,3,5,6,7,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> [5,3,4,6,7,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,[[.,.],[.,[[.,.],.]]]],.]]
=> [6,7,5,3,4,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> [5,6,7,3,4,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [6,7,4,3,5,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [6,7,3,4,5,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [7,4,3,5,6,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[.,[[[[.,.],.],.],[.,.]]],.]]
=> [7,3,4,5,6,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> [5,6,4,3,2,7,8,1] => ?
=> ? = 5
Description
The largest part of an integer partition.
Matching statistic: St001389
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> [.,.]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [2]
=> 2
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 2
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,1,1]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ?
=> ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [5,3,4,6,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [6,4,3,5,7,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [6,7,4,3,5,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [6,7,3,4,5,8,2,1] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,3,4,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [7,8,4,5,3,6,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [8,5,4,6,3,7,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [7,8,4,3,5,6,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ?
=> ? = 6
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [8,5,6,3,4,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 5
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [8,7,5,3,4,6,2,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [5,6,7,4,3,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[.,[[.,[[.,.],.]],.]]],.]]
=> [5,6,4,7,3,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,[[[.,[.,.]],.],.]]],.]]
=> [5,4,6,7,3,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],[.,.]]]],.]]
=> [7,5,4,6,3,2,8,1] => ?
=> ? = 6
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> [4,3,5,6,7,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> [5,3,4,6,7,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,[[.,.],[.,[[.,.],.]]]],.]]
=> [6,7,5,3,4,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> [5,6,7,3,4,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [6,7,4,3,5,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [6,7,3,4,5,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [7,4,3,5,6,2,8,1] => ?
=> ? = 5
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[.,[[[[.,.],.],.],[.,.]]],.]]
=> [7,3,4,5,6,2,8,1] => ?
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> [5,6,4,3,2,7,8,1] => ?
=> ? = 5
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000319
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 89%
Values
[1,0]
=> [.,.]
=> [1] => [1]
=> 0 = 1 - 1
[1,0,1,0]
=> [.,[.,.]]
=> [2,1] => [2]
=> 1 = 2 - 1
[1,1,0,0]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [3,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,[[.,.],.]],.]],.]]]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [5,3,4,6,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [.,[.,[[[[.,.],[[.,.],.]],.],.]]]
=> [5,6,3,4,7,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [6,4,3,5,7,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[[[.,.],.],.]],.]]]
=> [5,6,7,3,4,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[[.,.],[.,.]]],.]]]
=> [7,5,6,3,4,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [6,7,4,3,5,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [6,7,3,4,5,8,2,1] => ?
=> ? = 4 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,3,4,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> [8,5,6,4,3,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [7,8,4,5,3,6,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> [8,5,4,6,3,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [7,8,4,3,5,6,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => ?
=> ? = 6 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [8,5,6,3,4,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 5 - 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [8,7,5,3,4,6,2,1] => ?
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [5,6,7,4,3,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ?
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[.,[[.,[[.,.],.]],.]]],.]]
=> [5,6,4,7,3,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,[[[.,[.,.]],.],.]]],.]]
=> [5,4,6,7,3,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,[[.,[.,.]],[.,.]]]],.]]
=> [7,5,4,6,3,2,8,1] => ?
=> ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> [4,3,5,6,7,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> [5,3,4,6,7,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,[[.,.],[.,[[.,.],.]]]],.]]
=> [6,7,5,3,4,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> [5,6,7,3,4,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [6,7,4,3,5,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [6,7,3,4,5,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [7,4,3,5,6,2,8,1] => ?
=> ? = 5 - 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[.,[[[[.,.],.],.],[.,.]]],.]]
=> [7,3,4,5,6,2,8,1] => ?
=> ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> [5,6,4,3,2,7,8,1] => ?
=> ? = 5 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
The following 147 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000507The number of ascents of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St000097The order of the largest clique of the graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000098The chromatic number of a graph. St001581The achromatic number of a graph. St000925The number of topologically connected components of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000105The number of blocks in the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000172The Grundy number of a graph. St001029The size of the core of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000362The size of a minimal vertex cover of a graph. St000527The width of the poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001304The number of maximally independent sets of vertices of a graph. St000071The number of maximal chains in a poset. St000632The jump number of the poset. St000068The number of minimal elements in a poset. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000167The number of leaves of an ordered tree. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000996The number of exclusive left-to-right maxima of a permutation. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000245The number of ascents of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000702The number of weak deficiencies of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000308The height of the tree associated to a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000007The number of saliances of the permutation. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000332The positive inversions of an alternating sign matrix. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000542The number of left-to-right-minima of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000080The rank of the poset. St001153The number of blocks with even minimum in a set partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000061The number of nodes on the left branch of a binary tree. St000675The number of centered multitunnels of a Dyck path. St001480The number of simple summands of the module J^2/J^3. St001812The biclique partition number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000746The number of pairs with odd minimum in a perfect matching. St000822The Hadwiger number of the graph. St000331The number of upper interactions of a Dyck path. St000864The number of circled entries of the shifted recording tableau of a permutation. St001430The number of positive entries in a signed permutation. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001152The number of pairs with even minimum in a perfect matching. St000619The number of cyclic descents of a permutation. St000711The number of big exceedences of a permutation. St001668The number of points of the poset minus the width of the poset. St001330The hat guessing number of a graph. St001769The reflection length of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one.
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