Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000348
(load all 4 compositions to match this statistic)
Mapping
St000348: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 1
10 => 0
11 => 0
000 => 0
001 => 3
010 => 1
011 => 3
100 => 0
101 => 1
110 => 0
111 => 0
0000 => 0
0001 => 6
0010 => 3
0011 => 8
0100 => 1
0101 => 5
0110 => 3
0111 => 6
1000 => 0
1001 => 3
1010 => 1
1011 => 3
1100 => 0
1101 => 1
1110 => 0
1111 => 0
00000 => 0
00001 => 10
00010 => 6
00011 => 15
00100 => 3
00101 => 11
00110 => 8
00111 => 15
01000 => 1
01001 => 8
01010 => 5
01011 => 11
01100 => 3
01101 => 8
01110 => 6
01111 => 10
10000 => 0
10001 => 6
10010 => 3
10011 => 8
Description
The non-inversion sum of a binary word. A pair $a < b$ is an noninversion of a binary word $w = w_1 \cdots w_n$ if $w_a < w_b$. The non-inversion sum is given by $\sum(b-a)$ over all non-inversions of $w$.
Matching statistic: St000347
(load all 4 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 88%●distinct values known / distinct values provided: 71%
Values
0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 10 => 1
10 => 01 => 0
11 => 11 => 0
000 => 000 => 0
001 => 100 => 3
010 => 010 => 1
011 => 110 => 3
100 => 001 => 0
101 => 101 => 1
110 => 011 => 0
111 => 111 => 0
0000 => 0000 => 0
0001 => 1000 => 6
0010 => 0100 => 3
0011 => 1100 => 8
0100 => 0010 => 1
0101 => 1010 => 5
0110 => 0110 => 3
0111 => 1110 => 6
1000 => 0001 => 0
1001 => 1001 => 3
1010 => 0101 => 1
1011 => 1101 => 3
1100 => 0011 => 0
1101 => 1011 => 1
1110 => 0111 => 0
1111 => 1111 => 0
00000 => 00000 => 0
00001 => 10000 => 10
00010 => 01000 => 6
00011 => 11000 => 15
00100 => 00100 => 3
00101 => 10100 => 11
00110 => 01100 => 8
00111 => 11100 => 15
01000 => 00010 => 1
01001 => 10010 => 8
01010 => 01010 => 5
01011 => 11010 => 11
01100 => 00110 => 3
01101 => 10110 => 8
01110 => 01110 => 6
01111 => 11110 => 10
10000 => 00001 => 0
10001 => 10001 => 6
10010 => 01001 => 3
10011 => 11001 => 8
0000000001 => 1000000000 => ? = 45
0010000001 => 1000000100 => ? = 41
0010101101 => 1011010100 => ? = 73
0010101011 => 1101010100 => ? = 79
0010100111 => 1110010100 => ? = 86
0010011101 => 1011100100 => ? = 80
0010011011 => 1101100100 => ? = 86
0010010111 => 1110100100 => ? = 93
0010001111 => 1111000100 => ? = 101
0001110101 => 1010111000 => ? = 74
0001110011 => 1100111000 => ? = 80
0001101101 => 1011011000 => ? = 80
0001101011 => 1101011000 => ? = 86
0001100111 => 1110011000 => ? = 93
0001011101 => 1011101000 => ? = 87
0001011011 => 1101101000 => ? = 93
0001010111 => 1110101000 => ? = 100
0001001111 => 1111001000 => ? = 108
0000111101 => 1011110000 => ? = 95
0000111011 => 1101110000 => ? = 101
0000110111 => 1110110000 => ? = 108
0000101111 => 1111010000 => ? = 116
0000011111 => 1111100000 => ? = 125
0000000110 => 0110000000 => ? = 63
0000011110 => 0111100000 => ? = 90
0000010010 => 0100100000 => ? = 48
0001100110 => 0110011000 => ? = 62
0001111110 => 0111111000 => ? = 81
0001110010 => 0100111000 => ? = 51
0001001110 => 0111001000 => ? = 75
0001000010 => 0100001000 => ? = 37
0001011010 => 0101101000 => ? = 62
0010101010 => 0101010100 => ? = 50
0010101100 => 0011010100 => ? = 45
0010101110 => 0111010100 => ? = 68
0000000010 => 0100000000 => ? = 36
0000001110 => 0111000000 => ? = 81
0000111110 => 0111110000 => ? = 90
0000001010 => 0101000000 => ? = 55
0000010110 => 0110100000 => ? = 73
0000101110 => 0111010000 => ? = 82
0001011110 => 0111101000 => ? = 82
0001101110 => 0111011000 => ? = 75
0001110110 => 0110111000 => ? = 69
0001111010 => 0101111000 => ? = 64
0001111100 => 0011111000 => ? = 60
0000000101 => 1010000000 => ? = 71
0000001101 => 1011000000 => ? = 88
0000011101 => 1011100000 => ? = 96
0001111101 => 1011111000 => ? = 85
Description
The inversion sum of a binary word. A pair $a < b$ is an inversion of a binary word $w = w_1 \cdots w_n$ if $w_a = 1 > 0 = w_b$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
Matching statistic: St000869
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000869: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 39%
Values
0 => [2] => [[2],[]]
 => []
 => 0
1 => [1,1] => [[1,1],[]]
 => []
 => 0
00 => [3] => [[3],[]]
 => []
 => 0
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 1
10 => [1,2] => [[2,1],[]]
 => []
 => 0
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => 0
000 => [4] => [[4],[]]
 => []
 => 0
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 3
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 3
100 => [1,3] => [[3,1],[]]
 => []
 => 0
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => 0
0000 => [5] => [[5],[]]
 => []
 => 0
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 6
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 3
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 8
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 5
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 3
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 6
1000 => [1,4] => [[4,1],[]]
 => []
 => 0
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 3
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 3
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => 0
00000 => [6] => [[6],[]]
 => []
 => 0
00001 => [5,1] => [[5,5],[4]]
 => [4]
 => 10
00010 => [4,2] => [[5,4],[3]]
 => [3]
 => 6
00011 => [4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 15
00100 => [3,3] => [[5,3],[2]]
 => [2]
 => 3
00101 => [3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 11
00110 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 8
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 15
01000 => [2,4] => [[5,2],[1]]
 => [1]
 => 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 8
01010 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 5
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 11
01100 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 3
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 8
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 6
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 10
10000 => [1,5] => [[5,1],[]]
 => []
 => 0
10001 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 6
10010 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 3
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 8
0000010 => [6,2] => [[7,6],[5]]
 => ?
 => ? = 15
0000100 => [5,3] => [[7,5],[4]]
 => ?
 => ? = 10
0000101 => [5,2,1] => [[6,6,5],[5,4]]
 => ?
 => ? = 29
0000110 => [5,1,2] => [[6,5,5],[4,4]]
 => ?
 => ? = 24
0010000 => [3,5] => [[7,3],[2]]
 => ?
 => ? = 3
0011000 => [3,1,4] => [[6,3,3],[2,2]]
 => ?
 => ? = 8
0011110 => [3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
 => ?
 => ? = 24
0100000 => [2,6] => [[7,2],[1]]
 => ?
 => ? = 1
0100001 => [2,5,1] => [[6,6,2],[5,1]]
 => ?
 => ? = 17
0101000 => [2,2,4] => [[6,3,2],[2,1]]
 => ?
 => ? = 5
0110000 => [2,1,5] => [[6,2,2],[1,1]]
 => ?
 => ? = 3
0110001 => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
 => ?
 => ? = 15
0111000 => [2,1,1,4] => [[5,2,2,2],[1,1,1]]
 => ?
 => ? = 6
0111100 => [2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
 => ?
 => ? = 10
1000001 => [1,6,1] => [[6,6,1],[5]]
 => ?
 => ? = 15
1000010 => [1,5,2] => [[6,5,1],[4]]
 => ?
 => ? = 10
1000011 => [1,5,1,1] => [[5,5,5,1],[4,4]]
 => ?
 => ? = 24
1001000 => [1,3,4] => [[6,3,1],[2]]
 => ?
 => ? = 3
1001111 => [1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
 => ?
 => ? = 24
1010000 => [1,2,5] => [[6,2,1],[1]]
 => ?
 => ? = 1
1010001 => [1,2,4,1] => [[5,5,2,1],[4,1]]
 => ?
 => ? = 12
1011000 => [1,2,1,4] => [[5,2,2,1],[1,1]]
 => ?
 => ? = 3
1011100 => [1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
 => ?
 => ? = 6
1100001 => [1,1,5,1] => [[5,5,1,1],[4]]
 => ?
 => ? = 10
1100010 => [1,1,4,2] => [[5,4,1,1],[3]]
 => ?
 => ? = 6
1100011 => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
 => ?
 => ? = 15
1100111 => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
 => ?
 => ? = 15
1101000 => [1,1,2,4] => [[5,2,1,1],[1]]
 => ?
 => ? = 1
1101100 => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
 => ?
 => ? = 3
1110001 => [1,1,1,4,1] => [[4,4,1,1,1],[3]]
 => ?
 => ? = 6
1110011 => [1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
 => ?
 => ? = 8
1110100 => [1,1,1,2,3] => [[4,2,1,1,1],[1]]
 => ?
 => ? = 1
1111001 => [1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
 => ?
 => ? = 3
00000010 => [7,2] => [[8,7],[6]]
 => ?
 => ? = 21
00000100 => [6,3] => [[8,6],[5]]
 => ?
 => ? = 15
00000101 => [6,2,1] => [[7,7,6],[6,5]]
 => ?
 => ? = 41
00000110 => [6,1,2] => [[7,6,6],[5,5]]
 => ?
 => ? = 35
00001000 => [5,4] => [[8,5],[4]]
 => ?
 => ? = 10
00001001 => [5,3,1] => [[7,7,5],[6,4]]
 => ?
 => ? = 35
00001010 => [5,2,2] => [[7,6,5],[5,4]]
 => ?
 => ? = 29
00001011 => [5,2,1,1] => [[6,6,6,5],[5,5,4]]
 => ?
 => ? = 53
00001100 => [5,1,3] => [[7,5,5],[4,4]]
 => ?
 => ? = 24
00001101 => [5,1,2,1] => [[6,6,5,5],[5,4,4]]
 => ?
 => ? = 47
00001110 => [5,1,1,2] => [[6,5,5,5],[4,4,4]]
 => ?
 => ? = 42
00010000 => [4,5] => [[8,4],[3]]
 => ?
 => ? = 6
00010001 => [4,4,1] => [[7,7,4],[6,3]]
 => ?
 => ? = 30
00010010 => [4,3,2] => [[7,6,4],[5,3]]
 => ?
 => ? = 24
00010011 => [4,3,1,1] => [[6,6,6,4],[5,5,3]]
 => ?
 => ? = 47
00010100 => [4,2,3] => [[7,5,4],[4,3]]
 => ?
 => ? = 19
00010101 => [4,2,2,1] => [[6,6,5,4],[5,4,3]]
 => ?
 => ? = 41
Description
The sum of the hook lengths of an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the sum of all hook lengths of a partition.