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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000090
St000090: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 1
[2,1] => -1
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 0
[1,3] => 2
[2,1,1] => -1
[2,2] => 0
[3,1] => -2
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 0
[1,1,3] => 2
[1,2,1,1] => 0
[1,2,2] => 1
[1,3,1] => 0
[1,4] => 3
[2,1,1,1] => -1
[2,1,2] => 0
[2,2,1] => -1
[2,3] => 1
[3,1,1] => -2
[3,2] => -1
[4,1] => -3
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 0
[1,1,1,3] => 2
[1,1,2,1,1] => 0
[1,1,2,2] => 1
[1,1,3,1] => 0
[1,1,4] => 3
[1,2,1,1,1] => 0
[1,2,1,2] => 1
[1,2,2,1] => 0
[1,2,3] => 2
[1,3,1,1] => 0
[1,3,2] => 1
[1,4,1] => 0
[1,5] => 4
[2,1,1,1,1] => -1
[2,1,1,2] => 0
[2,1,2,1] => -1
Description
The variation of a composition.
Matching statistic: St000878
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> []
=> => ? = 0
[1,1] => [1,0,1,0]
=> [1]
=> 10 => 0
[2] => [1,1,0,0]
=> []
=> => ? = 0
[1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 1010 => 0
[1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 110 => 1
[2,1] => [1,1,0,0,1,0]
=> [2]
=> 100 => -1
[3] => [1,1,1,0,0,0]
=> []
=> => ? = 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101010 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => -1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => -2
[4] => [1,1,1,1,0,0,0,0]
=> []
=> => ? = 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10101010 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101010 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 10011010 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 111010 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 10100110 => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1100110 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 10001110 => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1010100 => -1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 110100 => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1001100 => -1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 11100 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 101000 => -2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 11000 => -1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => -3
[5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> => ? = 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 1010101010 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 110101010 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 1001101010 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 11101010 => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 1010011010 => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> 110011010 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 1000111010 => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1111010 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 1010100110 => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> 110100110 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1001100110 => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 11100110 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1010001110 => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 110001110 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 1000011110 => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 111110 => 4
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 101010100 => -1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> 11010100 => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 100110100 => -1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 1110100 => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> 101001100 => -1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 11001100 => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 100011100 => -1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 111100 => 2
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> => ? = 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> 11010101010 => ? = 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> 100110101010 => ? = 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> 1110101010 => ? = 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> 101001101010 => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> 11001101010 => ? = 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> 100011101010 => ? = 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> 101010011010 => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> 11010011010 => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> 100110011010 => ? = 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> 1110011010 => ? = 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> 101000111010 => ? = 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2,1]
=> 11000111010 => ? = 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> 100001111010 => ? = 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> 101010100110 => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1,1]
=> 11010100110 => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> 100110100110 => ? = 0
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,1,1]
=> 1110100110 => ? = 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> 101001100110 => ? = 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1,1]
=> 11001100110 => ? = 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1,1]
=> 100011100110 => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> 101010001110 => ? = 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1,1]
=> 11010001110 => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1,1]
=> 100110001110 => ? = 0
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,4,4,1,1,1]
=> 1110001110 => ? = 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> 101000011110 => ? = 0
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> 11000011110 => ? = 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> 10101010100 => ? = -1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> 10011010100 => ? = -1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> 10100110100 => ? = -1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2]
=> 10001110100 => ? = -1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> 10101001100 => ? = -1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2]
=> 10011001100 => ? = -1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2]
=> 10100011100 => ? = -1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> 1100011100 => ? = 0
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2]
=> 10000111100 => ? = -1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> 1010101000 => ? = -2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3]
=> 1001101000 => ? = -2
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3]
=> 1010011000 => ? = -2
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3]
=> 1000111000 => ? = -2
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> => ? = 0
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000145
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> []
=> ? = 0
[1,1] => [1,1] => [1,0,1,0]
=> [1]
=> 0
[2] => [2] => [1,1,0,0]
=> []
=> ? = 0
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,2] => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> -1
[3] => [3] => [1,1,1,0,0,0]
=> []
=> ? = 0
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 0
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> -1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> -2
[4] => [4] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 0
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 0
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 0
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 0
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 3
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> -1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 0
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> -1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> -2
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> -1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> -3
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 0
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 0
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 2
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 0
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 1
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? = 0
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 3
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 0
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 1
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 0
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 2
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? = 0
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 1
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 0
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 4
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = -1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 0
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = -1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 1
[2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = -1
[2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 0
[2,3,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = -1
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 2
[3,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? = -2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = -1
[3,2,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = -2
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 0
[4,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> -3
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> -2
[5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> -4
[6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 0
[1,1,1,1,1,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 1
[1,1,1,1,2,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 0
[1,1,1,1,3] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 2
[1,1,1,2,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 0
[1,1,1,2,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2]
=> ? = 1
[1,1,1,3,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 0
[1,1,1,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 3
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 0
[1,1,2,1,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2]
=> ? = 1
[1,1,2,2,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1,1]
=> ? = 0
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3]
=> ? = 2
[1,1,3,1,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2,1]
=> ? = 0
[1,1,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,2]
=> ? = 1
[1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> ? = 0
[1,1,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> ? = 4
[1,2,1,1,1,1] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 0
[1,2,1,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2]
=> ? = 1
[1,2,1,2,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1,1]
=> ? = 0
[1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3]
=> ? = 2
[1,2,2,1,1] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 0
[1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 5
[2,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> 3
[5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> -3
[6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> -5
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 $$p(7n+6) \equiv 0 \pmod 7.$$
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