searching the database
Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000878
(load all 81 compositions to match this statistic)
(load all 81 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: St000145
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
St000145: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> -1
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> -2
[2,1]
=> [2,1]
=> 0
[1,1,1]
=> [3]
=> 2
[4]
=> [1,1,1,1]
=> -3
[3,1]
=> [2,1,1]
=> -1
[2,2]
=> [2,2]
=> 0
[2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [4]
=> 3
[5]
=> [1,1,1,1,1]
=> -4
[4,1]
=> [2,1,1,1]
=> -2
[3,2]
=> [2,2,1]
=> -1
[3,1,1]
=> [3,1,1]
=> 0
[2,2,1]
=> [3,2]
=> 1
[2,1,1,1]
=> [4,1]
=> 2
[1,1,1,1,1]
=> [5]
=> 4
[6]
=> [1,1,1,1,1,1]
=> -5
[5,1]
=> [2,1,1,1,1]
=> -3
[4,2]
=> [2,2,1,1]
=> -2
[4,1,1]
=> [3,1,1,1]
=> -1
[3,3]
=> [2,2,2]
=> -1
[3,2,1]
=> [3,2,1]
=> 0
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 1
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 3
[1,1,1,1,1,1]
=> [6]
=> 5
[7]
=> [1,1,1,1,1,1,1]
=> -6
[6,1]
=> [2,1,1,1,1,1]
=> -4
[5,2]
=> [2,2,1,1,1]
=> -3
[5,1,1]
=> [3,1,1,1,1]
=> -2
[4,3]
=> [2,2,2,1]
=> -2
[4,2,1]
=> [3,2,1,1]
=> -1
[4,1,1,1]
=> [4,1,1,1]
=> 0
[3,3,1]
=> [3,2,2]
=> 0
[3,2,2]
=> [3,3,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> 1
[3,1,1,1,1]
=> [5,1,1]
=> 2
[2,2,2,1]
=> [4,3]
=> 2
[2,2,1,1,1]
=> [5,2]
=> 3
[2,1,1,1,1,1]
=> [6,1]
=> 4
[1,1,1,1,1,1,1]
=> [7]
=> 6
[8]
=> [1,1,1,1,1,1,1,1]
=> -7
[7,1]
=> [2,1,1,1,1,1,1]
=> -5
[6,2]
=> [2,2,1,1,1,1]
=> -4
[6,1,1]
=> [3,1,1,1,1,1]
=> -3
[5,3]
=> [2,2,2,1,1]
=> -3
[5,2,1]
=> [3,2,1,1,1]
=> -2
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> ? = -7
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> ? = -5
[7,4]
=> [2,2,2,2,1,1,1]
=> ? = -5
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = -3
[6,5]
=> [2,2,2,2,2,1]
=> ? = -4
[6,4,1]
=> [3,2,2,2,1,1]
=> ? = -3
[6,3,2]
=> [3,3,2,1,1,1]
=> ? = -3
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> ? = -1
[5,5,1]
=> [3,2,2,2,2]
=> ? = -2
[5,4,2]
=> [3,3,2,2,1]
=> ? = -2
[5,4,1,1]
=> [4,2,2,2,1]
=> ? = -1
[5,3,3]
=> [3,3,3,1,1]
=> ? = -2
[5,3,2,1]
=> [4,3,2,1,1]
=> ? = -1
[5,3,1,1,1]
=> [5,2,2,1,1]
=> ? = 0
[5,2,2,2]
=> [4,4,1,1,1]
=> ? = -1
[5,2,2,1,1]
=> [5,3,1,1,1]
=> ? = 0
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> ? = 1
[4,4,3]
=> [3,3,3,2]
=> ? = -1
[4,4,2,1]
=> [4,3,2,2]
=> ? = 0
[4,4,1,1,1]
=> [5,2,2,2]
=> ? = 1
[4,3,3,1]
=> [4,3,3,1]
=> ? = 0
[4,3,2,2]
=> [4,4,2,1]
=> ? = 0
[4,3,2,1,1]
=> [5,3,2,1]
=> ? = 1
[4,2,2,2,1]
=> [5,4,1,1]
=> ? = 1
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> ? = 3
[3,3,3,2]
=> [4,4,3]
=> ? = 1
[3,3,3,1,1]
=> [5,3,3]
=> ? = 2
[3,3,2,2,1]
=> [5,4,2]
=> ? = 2
[3,3,2,1,1,1]
=> [6,3,2]
=> ? = 3
[3,2,2,2,2]
=> [5,5,1]
=> ? = 2
[3,2,2,2,1,1]
=> [6,4,1]
=> ? = 3
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> ? = 5
[2,2,2,2,2,1]
=> [6,5]
=> ? = 4
[2,2,2,2,1,1,1]
=> [7,4]
=> ? = 5
[2,2,2,1,1,1,1,1]
=> [8,3]
=> ? = 6
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> ? = 7
[7,5]
=> [2,2,2,2,2,1,1]
=> ? = -5
[6,6]
=> [2,2,2,2,2,2]
=> ? = -4
[6,5,1]
=> [3,2,2,2,2,1]
=> ? = -3
[6,4,2]
=> [3,3,2,2,1,1]
=> ? = -3
[6,3,3]
=> [3,3,3,1,1,1]
=> ? = -3
[5,5,2]
=> [3,3,2,2,2]
=> ? = -2
[5,5,1,1]
=> [4,2,2,2,2]
=> ? = -1
[5,4,3]
=> [3,3,3,2,1]
=> ? = -2
[5,4,2,1]
=> [4,3,2,2,1]
=> ? = -1
[5,4,1,1,1]
=> [5,2,2,2,1]
=> ? = 0
[5,3,3,1]
=> [4,3,3,1,1]
=> ? = -1
[5,3,2,2]
=> [4,4,2,1,1]
=> ? = -1
[5,3,2,1,1]
=> [5,3,2,1,1]
=> ? = 0
[5,2,2,2,1]
=> [5,4,1,1,1]
=> ? = 0
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: 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: 18% ●values known / values provided: 18%●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: 18% ●values known / values provided: 18%●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
[6,1]
=> 10000010 => 00000011 => [6,2] => ? = -4
[5,2]
=> 1000100 => 0000011 => [5,2] => -3
[5,1,1]
=> 10000110 => 00000111 => [5,3] => ? = -2
[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] => ? = 0
[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] => ? = 2
[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] => ? = 4
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => [1,7] => ? = 6
[8]
=> 100000000 => 000000001 => [8,1] => ? = -7
[7,1]
=> 100000010 => 000000011 => [7,2] => ? = -5
[6,2]
=> 10000100 => 00000011 => [6,2] => ? = -4
[6,1,1]
=> 100000110 => 000000111 => [6,3] => ? = -3
[5,3]
=> 1001000 => 0000011 => [5,2] => -3
[5,2,1]
=> 10001010 => 00001011 => [4,1,1,2] => ? = -2
[5,1,1,1]
=> 100001110 => 000001111 => [5,4] => ? = -1
[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] => ? = 0
[4,1,1,1,1]
=> 100011110 => 000011111 => [4,5] => ? = 1
[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] => ? = 2
[3,1,1,1,1,1]
=> 100111110 => 000111111 => [3,6] => ? = 3
[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] => ? = 4
[2,1,1,1,1,1,1]
=> 101111110 => 001111111 => [2,7] => ? = 5
[1,1,1,1,1,1,1,1]
=> 111111110 => 011111111 => [1,8] => ? = 7
[9]
=> 1000000000 => 0000000001 => [9,1] => ? = -8
[8,1]
=> 1000000010 => 0000000011 => [8,2] => ? = -6
[7,2]
=> 100000100 => 000000011 => [7,2] => ? = -5
[7,1,1]
=> 1000000110 => 0000000111 => [7,3] => ? = -4
[6,3]
=> 10001000 => 00000011 => [6,2] => ? = -4
[6,2,1]
=> 100001010 => 000001011 => [5,1,1,2] => ? = -3
[6,1,1,1]
=> 1000001110 => 0000001111 => [6,4] => ? = -2
[5,4]
=> 1010000 => 0000011 => [5,2] => -3
[5,3,1]
=> 10010010 => 00010011 => [3,1,2,2] => ? = -2
[5,2,2]
=> 10001100 => 00000111 => [5,3] => ? = -2
[5,2,1,1]
=> 100010110 => 000010111 => [4,1,1,3] => ? = -1
[5,1,1,1,1]
=> 1000011110 => 0000011111 => [5,5] => ? = 0
[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] => ? = 0
[4,2,2,1]
=> 10011010 => 00011011 => [3,2,1,2] => ? = 0
[4,2,1,1,1]
=> 100101110 => 000101111 => [3,1,1,4] => ? = 1
[4,1,1,1,1,1]
=> 1000111110 => 0000111111 => [4,6] => ? = 2
[3,3,3]
=> 111000 => 000111 => [3,3] => 0
[3,3,1,1,1]
=> 11001110 => 00011111 => [3,5] => ? = 2
[3,2,2,1,1]
=> 10110110 => 00110111 => [2,2,1,3] => ? = 2
[3,2,1,1,1,1]
=> 101011110 => 001011111 => [2,1,1,5] => ? = 3
[3,1,1,1,1,1,1]
=> 1001111110 => 0001111111 => [3,7] => ? = 4
[2,2,2,1,1,1]
=> 11101110 => 00111111 => [2,6] => ? = 4
[2,2,1,1,1,1,1]
=> 110111110 => 001111111 => [2,7] => ? = 5
[2,1,1,1,1,1,1,1]
=> 1011111110 => 0011111111 => [2,8] => ? = 6
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => 0111111111 => [1,9] => ? = 8
[10]
=> 10000000000 => 00000000001 => [10,1] => ? = -9
[9,1]
=> 10000000010 => 00000000011 => [9,2] => ? = -7
[8,2]
=> 1000000100 => 0000000011 => [8,2] => ? = -6
[8,1,1]
=> 10000000110 => 00000000111 => [8,3] => ? = -5
[7,3]
=> 100001000 => 000000011 => [7,2] => ? = -5
[7,2,1]
=> 1000001010 => 0000001011 => [6,1,1,2] => ? = -4
[7,1,1,1]
=> 10000001110 => 00000001111 => [7,4] => ? = -3
Description
The variation of a composition.
Matching statistic: St000894
(load all 3 compositions to match this statistic)
(load all 3 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: 14% ●values known / values provided: 14%●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: 14% ●values known / values provided: 14%●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 + 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]]
=> ? = -4 + 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]]
=> ? = -2 + 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]]
=> ? = 0 + 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]]
=> ? = 2 + 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]]
=> ? = 4 + 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 + 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 + 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]]
=> ? = -5 + 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]]
=> ? = -4 + 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]]
=> ? = -3 + 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]]
=> ? = -2 + 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]]
=> ? = -1 + 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]]
=> ? = -1 + 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]]
=> ? = 0 + 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]]
=> ? = 1 + 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]]
=> ? = 2 + 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]]
=> ? = 3 + 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]]
=> ? = 4 + 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]]
=> ? = 5 + 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 + 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 + 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]]
=> ? = -6 + 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]]
=> ? = -5 + 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]]
=> ? = -4 + 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]]
=> ? = -4 + 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]]
=> ? = -3 + 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]]
=> ? = -2 + 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]]
=> ? = -2 + 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]]
=> ? = -2 + 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]]
=> ? = -1 + 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]]
=> ? = 0 + 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]]
=> ? = -1 + 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]]
=> ? = 0 + 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]]
=> ? = 0 + 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]]
=> ? = 1 + 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]]
=> ? = 2 + 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]]
=> ? = 2 + 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]]
=> ? = 1 + 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]]
=> ? = 2 + 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]]
=> ? = 3 + 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]]
=> ? = 4 + 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]]
=> ? = 4 + 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]]
=> ? = 5 + 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]]
=> ? = 6 + 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 + 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 + 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]]
=> ? = -7 + 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]]
=> ? = -6 + 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.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!