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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000145
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> -1
[3]
=> 2
[2,1]
=> 0
[1,1,1]
=> -2
[4]
=> 3
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> -1
[1,1,1,1]
=> -3
[5]
=> 4
[4,1]
=> 2
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> -1
[2,1,1,1]
=> -2
[1,1,1,1,1]
=> -4
[6]
=> 5
[5,1]
=> 3
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 0
[3,1,1,1]
=> -1
[2,2,2]
=> -1
[2,2,1,1]
=> -2
[2,1,1,1,1]
=> -3
[1,1,1,1,1,1]
=> -5
[7]
=> 6
[6,1]
=> 4
[5,2]
=> 3
[5,1,1]
=> 2
[4,3]
=> 2
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 0
[3,2,2]
=> 0
[3,2,1,1]
=> -1
[3,1,1,1,1]
=> -2
[2,2,2,1]
=> -2
[2,2,1,1,1]
=> -3
[2,1,1,1,1,1]
=> -4
[1,1,1,1,1,1,1]
=> -6
[8]
=> 7
[7,1]
=> 5
[6,2]
=> 4
[6,1,1]
=> 3
[5,3]
=> 3
[5,2,1]
=> 2
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000878
(load all 88 compositions to match this statistic)
(load all 88 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 0
[2]
=> 100 => -1
[1,1]
=> 110 => 1
[3]
=> 1000 => -2
[2,1]
=> 1010 => 0
[1,1,1]
=> 1110 => 2
[4]
=> 10000 => -3
[3,1]
=> 10010 => -1
[2,2]
=> 1100 => 0
[2,1,1]
=> 10110 => 1
[1,1,1,1]
=> 11110 => 3
[5]
=> 100000 => -4
[4,1]
=> 100010 => -2
[3,2]
=> 10100 => -1
[3,1,1]
=> 100110 => 0
[2,2,1]
=> 11010 => 1
[2,1,1,1]
=> 101110 => 2
[1,1,1,1,1]
=> 111110 => 4
[6]
=> 1000000 => -5
[5,1]
=> 1000010 => -3
[4,2]
=> 100100 => -2
[4,1,1]
=> 1000110 => -1
[3,3]
=> 11000 => -1
[3,2,1]
=> 101010 => 0
[3,1,1,1]
=> 1001110 => 1
[2,2,2]
=> 11100 => 1
[2,2,1,1]
=> 110110 => 2
[2,1,1,1,1]
=> 1011110 => 3
[1,1,1,1,1,1]
=> 1111110 => 5
[7]
=> 10000000 => -6
[6,1]
=> 10000010 => -4
[5,2]
=> 1000100 => -3
[5,1,1]
=> 10000110 => -2
[4,3]
=> 101000 => -2
[4,2,1]
=> 1001010 => -1
[4,1,1,1]
=> 10001110 => 0
[3,3,1]
=> 110010 => 0
[3,2,2]
=> 101100 => 0
[3,2,1,1]
=> 1010110 => 1
[3,1,1,1,1]
=> 10011110 => 2
[2,2,2,1]
=> 111010 => 2
[2,2,1,1,1]
=> 1101110 => 3
[2,1,1,1,1,1]
=> 10111110 => 4
[1,1,1,1,1,1,1]
=> 11111110 => 6
[8]
=> 100000000 => -7
[7,1]
=> 100000010 => -5
[6,2]
=> 10000100 => -4
[6,1,1]
=> 100000110 => -3
[5,3]
=> 1001000 => -3
[5,2,1]
=> 10001010 => -2
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000090
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 58%
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000090: Integer compositions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 58%
Values
[1]
=> 10 => 01 => [1,1] => 0
[2]
=> 100 => 001 => [2,1] => -1
[1,1]
=> 110 => 011 => [1,2] => 1
[3]
=> 1000 => 0001 => [3,1] => -2
[2,1]
=> 1010 => 0011 => [2,2] => 0
[1,1,1]
=> 1110 => 0111 => [1,3] => 2
[4]
=> 10000 => 00001 => [4,1] => -3
[3,1]
=> 10010 => 00011 => [3,2] => -1
[2,2]
=> 1100 => 0011 => [2,2] => 0
[2,1,1]
=> 10110 => 00111 => [2,3] => 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 3
[5]
=> 100000 => 000001 => [5,1] => -4
[4,1]
=> 100010 => 000011 => [4,2] => -2
[3,2]
=> 10100 => 00011 => [3,2] => -1
[3,1,1]
=> 100110 => 000111 => [3,3] => 0
[2,2,1]
=> 11010 => 00111 => [2,3] => 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 2
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 4
[6]
=> 1000000 => 0000001 => [6,1] => -5
[5,1]
=> 1000010 => 0000011 => [5,2] => -3
[4,2]
=> 100100 => 000011 => [4,2] => -2
[4,1,1]
=> 1000110 => 0000111 => [4,3] => -1
[3,3]
=> 11000 => 00011 => [3,2] => -1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 0
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 2
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 3
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 5
[7]
=> 10000000 => 00000001 => [7,1] => ? ∊ {-6,-4,-2,0,2,4,6}
[6,1]
=> 10000010 => 00000011 => [6,2] => ? ∊ {-6,-4,-2,0,2,4,6}
[5,2]
=> 1000100 => 0000011 => [5,2] => -3
[5,1,1]
=> 10000110 => 00000111 => [5,3] => ? ∊ {-6,-4,-2,0,2,4,6}
[4,3]
=> 101000 => 000011 => [4,2] => -2
[4,2,1]
=> 1001010 => 0001011 => [3,1,1,2] => -1
[4,1,1,1]
=> 10001110 => 00001111 => [4,4] => ? ∊ {-6,-4,-2,0,2,4,6}
[3,3,1]
=> 110010 => 000111 => [3,3] => 0
[3,2,2]
=> 101100 => 000111 => [3,3] => 0
[3,2,1,1]
=> 1010110 => 0010111 => [2,1,1,3] => 1
[3,1,1,1,1]
=> 10011110 => 00011111 => [3,5] => ? ∊ {-6,-4,-2,0,2,4,6}
[2,2,2,1]
=> 111010 => 001111 => [2,4] => 2
[2,2,1,1,1]
=> 1101110 => 0011111 => [2,5] => 3
[2,1,1,1,1,1]
=> 10111110 => 00111111 => [2,6] => ? ∊ {-6,-4,-2,0,2,4,6}
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => [1,7] => ? ∊ {-6,-4,-2,0,2,4,6}
[8]
=> 100000000 => 000000001 => [8,1] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[7,1]
=> 100000010 => 000000011 => [7,2] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[6,2]
=> 10000100 => 00000011 => [6,2] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[6,1,1]
=> 100000110 => 000000111 => [6,3] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[5,3]
=> 1001000 => 0000011 => [5,2] => -3
[5,2,1]
=> 10001010 => 00001011 => [4,1,1,2] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[5,1,1,1]
=> 100001110 => 000001111 => [5,4] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[4,4]
=> 110000 => 000011 => [4,2] => -2
[4,3,1]
=> 1010010 => 0001011 => [3,1,1,2] => -1
[4,2,2]
=> 1001100 => 0000111 => [4,3] => -1
[4,2,1,1]
=> 10010110 => 00010111 => [3,1,1,3] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[4,1,1,1,1]
=> 100011110 => 000011111 => [4,5] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[3,3,2]
=> 110100 => 000111 => [3,3] => 0
[3,3,1,1]
=> 1100110 => 0001111 => [3,4] => 1
[3,2,2,1]
=> 1011010 => 0010111 => [2,1,1,3] => 1
[3,2,1,1,1]
=> 10101110 => 00101111 => [2,1,1,4] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[3,1,1,1,1,1]
=> 100111110 => 000111111 => [3,6] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[2,2,2,2]
=> 111100 => 001111 => [2,4] => 2
[2,2,2,1,1]
=> 1110110 => 0011111 => [2,5] => 3
[2,2,1,1,1,1]
=> 11011110 => 00111111 => [2,6] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[2,1,1,1,1,1,1]
=> 101111110 => 001111111 => [2,7] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[1,1,1,1,1,1,1,1]
=> 111111110 => 011111111 => [1,8] => ? ∊ {-7,-5,-4,-3,-2,-1,0,1,2,3,4,5,7}
[9]
=> 1000000000 => 0000000001 => [9,1] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[8,1]
=> 1000000010 => 0000000011 => [8,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[7,2]
=> 100000100 => 000000011 => [7,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[7,1,1]
=> 1000000110 => 0000000111 => [7,3] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[6,3]
=> 10001000 => 00000011 => [6,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[6,2,1]
=> 100001010 => 000001011 => [5,1,1,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[6,1,1,1]
=> 1000001110 => 0000001111 => [6,4] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[5,4]
=> 1010000 => 0000011 => [5,2] => -3
[5,3,1]
=> 10010010 => 00010011 => [3,1,2,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[5,2,2]
=> 10001100 => 00000111 => [5,3] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[5,2,1,1]
=> 100010110 => 000010111 => [4,1,1,3] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[5,1,1,1,1]
=> 1000011110 => 0000011111 => [5,5] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[4,4,1]
=> 1100010 => 0000111 => [4,3] => -1
[4,3,2]
=> 1010100 => 0001011 => [3,1,1,2] => -1
[4,3,1,1]
=> 10100110 => 00011011 => [3,2,1,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[4,2,2,1]
=> 10011010 => 00011011 => [3,2,1,2] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[4,2,1,1,1]
=> 100101110 => 000101111 => [3,1,1,4] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[4,1,1,1,1,1]
=> 1000111110 => 0000111111 => [4,6] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[3,3,3]
=> 111000 => 000111 => [3,3] => 0
[3,3,1,1,1]
=> 11001110 => 00011111 => [3,5] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[3,2,2,1,1]
=> 10110110 => 00110111 => [2,2,1,3] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[3,2,1,1,1,1]
=> 101011110 => 001011111 => [2,1,1,5] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[3,1,1,1,1,1,1]
=> 1001111110 => 0001111111 => [3,7] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[2,2,2,1,1,1]
=> 11101110 => 00111111 => [2,6] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[2,2,1,1,1,1,1]
=> 110111110 => 001111111 => [2,7] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0011111111 => [2,8] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0111111111 => [1,9] => ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,0,0,0,1,2,2,2,3,4,4,5,6,8}
[10]
=> 10000000000 => 00000000001 => [10,1] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
[9,1]
=> 10000000010 => 00000000011 => [9,2] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
[8,2]
=> 1000000100 => 0000000011 => [8,2] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
[8,1,1]
=> 10000000110 => 00000000111 => [8,3] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
[7,3]
=> 100001000 => 000000011 => [7,2] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
[7,2,1]
=> 1000001010 => 0000001011 => [6,1,1,2] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
[7,1,1,1]
=> 10000001110 => 00000001111 => [7,4] => ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,9}
Description
The variation of a composition.
Matching statistic: St000894
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 58%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 58%
Values
[1]
=> [1,0]
=> [1,0]
=> [[1]]
=> ? = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0 = -1 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> -1 = -2 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> -2 = -3 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 0 = -1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 3 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> -3 = -4 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> -1 = -2 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0 = -1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5 = 4 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> -4 = -5 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> -2 = -3 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> -1 = -2 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 0 = -1 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 0 = -1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 6 = 5 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> -2 = -3 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> -1 = -2 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 0 = -1 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 0 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3 = 2 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-6,-4,-2,0,2,4,6} + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> -2 = -3 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> -1 = -2 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 0 = -1 + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 3 = 2 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,5,7} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> -2 = -3 + 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 0 = -1 + 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> 2 = 1 + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> 4 = 3 + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-2,-1,-1,0,0,0,1,1,2,2,2,3,4,4,5,6,8} + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,5,5,6,7,9} + 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,5,5,6,7,9} + 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {-9,-7,-6,-5,-5,-4,-4,-3,-3,-3,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,5,5,6,7,9} + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> -2 = -3 + 1
Description
The trace of an alternating sign matrix.
Matching statistic: St000259
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 37%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 37%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 0
[2]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> ? = -1
[3]
=> [1,0,1,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {-2,0}
[2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {-2,0}
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-3,-1,0,1}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {-3,-1,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-3,-1,0,1}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-3,-1,0,1}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {-4,-2,-1,0,1}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {-4,-2,-1,0,1}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-4,-2,-1,0,1}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-4,-2,-1,0,1}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {-4,-2,-1,0,1}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 5
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ? ∊ {-5,-3,-2,-1,-1,0,1,1,2}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 6
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,1,2}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 5
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5,8] => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => ([(1,2),(3,6),(4,5),(5,7),(6,7)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6,8] => ([(1,4),(2,3),(5,7),(6,7)],8)
=> ? ∊ {-7,-5,-4,-3,-3,-2,-2,-1,-1,-1,0,0,1,1,1,2,2,3,7}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
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