Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000878
(load all 71 compositions to match this statistic)
Mapping
St000878: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => -1
1 => 1
00 => -2
01 => 0
10 => 0
11 => 2
000 => -3
001 => -1
010 => -1
011 => 1
100 => -1
101 => 1
110 => 1
111 => 3
0000 => -4
0001 => -2
0010 => -2
0011 => 0
0100 => -2
0101 => 0
0110 => 0
0111 => 2
1000 => -2
1001 => 0
1010 => 0
1011 => 2
1100 => 0
1101 => 2
1110 => 2
1111 => 4
00000 => -5
00001 => -3
00010 => -3
00011 => -1
00100 => -3
00101 => -1
00110 => -1
00111 => 1
01000 => -3
01001 => -1
01010 => -1
01011 => 1
01100 => -1
01101 => 1
01110 => 1
01111 => 3
10000 => -3
10001 => -1
10010 => -1
10011 => 1
Description
The number of ones minus the number of zeros of a binary word.
Matching statistic: St000145
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => [2]
 => 1
1 => [1,1] => [[1,1],[]]
 => [1,1]
 => -1
00 => [3] => [[3],[]]
 => [3]
 => 2
01 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 0
10 => [1,2] => [[2,1],[]]
 => [2,1]
 => 0
11 => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => -2
000 => [4] => [[4],[]]
 => [4]
 => 3
001 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => -1
100 => [1,3] => [[3,1],[]]
 => [3,1]
 => 1
101 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => -1
110 => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => -1
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => -3
0000 => [5] => [[5],[]]
 => [5]
 => 4
0001 => [4,1] => [[4,4],[3]]
 => [4,4]
 => 2
0010 => [3,2] => [[4,3],[2]]
 => [4,3]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 0
0100 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 0
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => -2
1000 => [1,4] => [[4,1],[]]
 => [4,1]
 => 2
1001 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 0
1010 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => -2
1100 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => -2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => -2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => -4
00000 => [6] => [[6],[]]
 => [6]
 => 5
00001 => [5,1] => [[5,5],[4]]
 => [5,5]
 => 3
00010 => [4,2] => [[5,4],[3]]
 => [5,4]
 => 3
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [4,4,4]
 => ? ∊ {-1,-1,1,1}
00100 => [3,3] => [[5,3],[2]]
 => [5,3]
 => 3
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [4,4,3]
 => ? ∊ {-1,-1,1,1}
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [3,3,3,3]
 => ? ∊ {-1,-1,1,1}
01000 => [2,4] => [[5,2],[1]]
 => [5,2]
 => 3
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [4,4,2]
 => 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [3,3,3,2]
 => ? ∊ {-1,-1,1,1}
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [3,3,2,2]
 => -1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => -1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [2,2,2,2,2]
 => -3
10000 => [1,5] => [[5,1],[]]
 => [5,1]
 => 3
10001 => [1,4,1] => [[4,4,1],[3]]
 => [4,4,1]
 => 1
10010 => [1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [3,3,3,1]
 => -1
10100 => [1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [3,3,2,1]
 => -1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => -1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [2,2,2,2,1]
 => -3
000001 => [6,1] => [[6,6],[5]]
 => [6,6]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
000010 => [5,2] => [[6,5],[4]]
 => [6,5]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
000011 => [5,1,1] => [[5,5,5],[4,4]]
 => [5,5,5]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
000101 => [4,2,1] => [[5,5,4],[4,3]]
 => [5,5,4]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
000110 => [4,1,2] => [[5,4,4],[3,3]]
 => [5,4,4]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
 => [4,4,4,4]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001001 => [3,3,1] => [[5,5,3],[4,2]]
 => [5,5,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001010 => [3,2,2] => [[5,4,3],[3,2]]
 => [5,4,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => [4,4,4,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001100 => [3,1,3] => [[5,3,3],[2,2]]
 => [5,3,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => [4,4,3,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [4,3,3,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
 => [3,3,3,3,3]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
010001 => [2,4,1] => [[5,5,2],[4,1]]
 => [5,5,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
010010 => [2,3,2] => [[5,4,2],[3,1]]
 => [5,4,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => [4,4,4,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]]
 => [4,4,3,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [4,3,3,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => [3,3,3,3,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => [4,4,2,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [4,3,2,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
 => [3,3,3,2,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => [3,3,2,2,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [3,2,2,2,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
 => [2,2,2,2,2,2]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
100001 => [1,5,1] => [[5,5,1],[4]]
 => [5,5,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]]
 => [4,4,4,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]]
 => [4,4,3,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]]
 => [4,3,3,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
100111 => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
 => [3,3,3,3,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]]
 => [4,4,2,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
101011 => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
 => [3,3,3,2,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
101101 => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
 => [3,3,2,2,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
101111 => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
 => [2,2,2,2,2,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => [3,3,3,1,1]
 => ? ∊ {-4,-4,-2,-2,-2,-2,-2,-2,-2,-2,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,2,2,4,4}
0000001 => [7,1] => [[7,7],[6]]
 => [7,7]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0000011 => [6,1,1] => [[6,6,6],[5,5]]
 => [6,6,6]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0000111 => [5,1,1,1] => [[5,5,5,5],[4,4,4]]
 => [5,5,5,5]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0001000 => [4,4] => [[7,4],[3]]
 => [7,4]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0001001 => [4,3,1] => [[6,6,4],[5,3]]
 => [6,6,4]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0001010 => [4,2,2] => [[6,5,4],[4,3]]
 => [6,5,4]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
0001011 => [4,2,1,1] => [[5,5,5,4],[4,4,3]]
 => [5,5,5,4]
 => ? ∊ {-5,-5,-5,-5,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-3,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5}
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 $$p(7n+6) \equiv 0 \pmod 7.$$