Your data matches 2 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●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
00100 => [3,3] => [[5,3],[2]]
 => [5,3]
 => 3
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 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
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
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 113 compositions to match this statistic)
Mapping
Mp00105: Binary words —complement⟶ Binary words
St000878: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
0 => 1 => 1
1 => 0 => -1
00 => 11 => 2
01 => 10 => 0
10 => 01 => 0
11 => 00 => -2
000 => 111 => 3
001 => 110 => 1
010 => 101 => 1
011 => 100 => -1
100 => 011 => 1
101 => 010 => -1
110 => 001 => -1
111 => 000 => -3
0000 => 1111 => 4
0001 => 1110 => 2
0010 => 1101 => 2
0011 => 1100 => 0
0100 => 1011 => 2
0101 => 1010 => 0
0110 => 1001 => 0
0111 => 1000 => -2
1000 => 0111 => 2
1001 => 0110 => 0
1010 => 0101 => 0
1011 => 0100 => -2
1100 => 0011 => 0
1101 => 0010 => -2
1110 => 0001 => -2
1111 => 0000 => -4
00000 => 11111 => 5
00001 => 11110 => 3
00010 => 11101 => 3
00100 => 11011 => 3
00110 => 11001 => 1
01000 => 10111 => 3
01001 => 10110 => 1
01010 => 10101 => 1
01100 => 10011 => 1
01101 => 10010 => -1
01110 => 10001 => -1
01111 => 10000 => -3
10000 => 01111 => 3
10001 => 01110 => 1
10010 => 01101 => 1
10011 => 01100 => -1
10100 => 01011 => 1
10101 => 01010 => -1
10110 => 01001 => -1
10111 => 01000 => -3
=> => ? = 0
Description
The number of ones minus the number of zeros of a binary word.