Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000348
(load all 2 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000348: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 10 => 01 => 1
[2]
 => 100 => 011 => 3
[1,1]
 => 110 => 001 => 3
[3]
 => 1000 => 0111 => 6
[2,1]
 => 1010 => 0101 => 5
[1,1,1]
 => 1110 => 0001 => 6
[4]
 => 10000 => 01111 => 10
[3,1]
 => 10010 => 01101 => 8
[2,2]
 => 1100 => 0011 => 8
[2,1,1]
 => 10110 => 01001 => 8
[1,1,1,1]
 => 11110 => 00001 => 10
[5]
 => 100000 => 011111 => 15
[4,1]
 => 100010 => 011101 => 12
[3,2]
 => 10100 => 01011 => 11
[3,1,1]
 => 100110 => 011001 => 11
[2,2,1]
 => 11010 => 00101 => 11
[2,1,1,1]
 => 101110 => 010001 => 12
[1,1,1,1,1]
 => 111110 => 000001 => 15
[6]
 => 1000000 => 0111111 => 21
[5,1]
 => 1000010 => 0111101 => 17
[4,2]
 => 100100 => 011011 => 15
[4,1,1]
 => 1000110 => 0111001 => 15
[3,3]
 => 11000 => 00111 => 15
[3,2,1]
 => 101010 => 010101 => 14
[3,1,1,1]
 => 1001110 => 0110001 => 15
[2,2,2]
 => 11100 => 00011 => 15
[2,2,1,1]
 => 110110 => 001001 => 15
[2,1,1,1,1]
 => 1011110 => 0100001 => 17
[1,1,1,1,1,1]
 => 1111110 => 0000001 => 21
[7]
 => 10000000 => 01111111 => 28
[6,1]
 => 10000010 => 01111101 => 23
[5,2]
 => 1000100 => 0111011 => 20
[5,1,1]
 => 10000110 => 01111001 => 20
[4,3]
 => 101000 => 010111 => 19
[4,2,1]
 => 1001010 => 0110101 => 18
[4,1,1,1]
 => 10001110 => 01110001 => 19
[3,3,1]
 => 110010 => 001101 => 18
[3,2,2]
 => 101100 => 010011 => 18
[3,2,1,1]
 => 1010110 => 0101001 => 18
[3,1,1,1,1]
 => 10011110 => 01100001 => 20
[2,2,2,1]
 => 111010 => 000101 => 19
[2,2,1,1,1]
 => 1101110 => 0010001 => 20
[2,1,1,1,1,1]
 => 10111110 => 01000001 => 23
[1,1,1,1,1,1,1]
 => 11111110 => 00000001 => 28
[8]
 => 100000000 => 011111111 => 36
[7,1]
 => 100000010 => 011111101 => 30
[6,2]
 => 10000100 => 01111011 => 26
[6,1,1]
 => 100000110 => 011111001 => 26
[5,3]
 => 1001000 => 0110111 => 24
[5,2,1]
 => 10001010 => 01110101 => 23
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: St000869
(load all 3 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000869: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 96%
Values
[1]
 => [1]
 => 1
[2]
 => [1,1]
 => 3
[1,1]
 => [2]
 => 3
[3]
 => [1,1,1]
 => 6
[2,1]
 => [2,1]
 => 5
[1,1,1]
 => [3]
 => 6
[4]
 => [1,1,1,1]
 => 10
[3,1]
 => [2,1,1]
 => 8
[2,2]
 => [2,2]
 => 8
[2,1,1]
 => [3,1]
 => 8
[1,1,1,1]
 => [4]
 => 10
[5]
 => [1,1,1,1,1]
 => 15
[4,1]
 => [2,1,1,1]
 => 12
[3,2]
 => [2,2,1]
 => 11
[3,1,1]
 => [3,1,1]
 => 11
[2,2,1]
 => [3,2]
 => 11
[2,1,1,1]
 => [4,1]
 => 12
[1,1,1,1,1]
 => [5]
 => 15
[6]
 => [1,1,1,1,1,1]
 => 21
[5,1]
 => [2,1,1,1,1]
 => 17
[4,2]
 => [2,2,1,1]
 => 15
[4,1,1]
 => [3,1,1,1]
 => 15
[3,3]
 => [2,2,2]
 => 15
[3,2,1]
 => [3,2,1]
 => 14
[3,1,1,1]
 => [4,1,1]
 => 15
[2,2,2]
 => [3,3]
 => 15
[2,2,1,1]
 => [4,2]
 => 15
[2,1,1,1,1]
 => [5,1]
 => 17
[1,1,1,1,1,1]
 => [6]
 => 21
[7]
 => [1,1,1,1,1,1,1]
 => 28
[6,1]
 => [2,1,1,1,1,1]
 => 23
[5,2]
 => [2,2,1,1,1]
 => 20
[5,1,1]
 => [3,1,1,1,1]
 => 20
[4,3]
 => [2,2,2,1]
 => 19
[4,2,1]
 => [3,2,1,1]
 => 18
[4,1,1,1]
 => [4,1,1,1]
 => 19
[3,3,1]
 => [3,2,2]
 => 18
[3,2,2]
 => [3,3,1]
 => 18
[3,2,1,1]
 => [4,2,1]
 => 18
[3,1,1,1,1]
 => [5,1,1]
 => 20
[2,2,2,1]
 => [4,3]
 => 19
[2,2,1,1,1]
 => [5,2]
 => 20
[2,1,1,1,1,1]
 => [6,1]
 => 23
[1,1,1,1,1,1,1]
 => [7]
 => 28
[8]
 => [1,1,1,1,1,1,1,1]
 => 36
[7,1]
 => [2,1,1,1,1,1,1]
 => 30
[6,2]
 => [2,2,1,1,1,1]
 => 26
[6,1,1]
 => [3,1,1,1,1,1]
 => 26
[5,3]
 => [2,2,2,1,1]
 => 24
[5,2,1]
 => [3,2,1,1,1]
 => 23
[5,5,2,1]
 => [4,3,2,2,2]
 => ? = 46
[5,5,3,1]
 => [4,3,3,2,2]
 => ? = 51
[5,5,2,2]
 => [4,4,2,2,2]
 => ? = 51
[5,4,4,1]
 => [4,3,3,3,1]
 => ? = 51
[4,4,3,1,1,1]
 => [6,3,3,2]
 => ? = 51
[4,4,2,2,2]
 => [5,5,2,2]
 => ? = 51
[4,3,3,2,2]
 => [5,5,3,1]
 => ? = 51
[5,5,3,2]
 => [4,4,3,2,2]
 => ? = 56
[5,4,4,2]
 => [4,4,3,3,1]
 => ? = 56
[5,4,3,3]
 => [4,4,4,2,1]
 => ? = 56
[4,4,3,3,1]
 => [5,4,4,2]
 => ? = 56
[4,4,3,2,2]
 => [5,5,3,2]
 => ? = 56
[4,4,3,2,1,1]
 => [6,4,3,2]
 => ? = 56
[4,3,3,3,2]
 => [5,5,4,1]
 => ? = 57
[3,3,3,3,1,1,1]
 => [7,4,4]
 => ? = 60
[3,3,2,2,2,2,1]
 => [7,6,2]
 => ? = 62
[6,5,5]
 => [3,3,3,3,3,1]
 => ? = 66
[5,4,4,3]
 => [4,4,4,3,1]
 => ? = 62
[4,4,3,3,2]
 => [5,5,4,2]
 => ? = 62
[4,4,3,3,1,1]
 => [6,4,4,2]
 => ? = 62
[4,4,3,2,2,1]
 => [6,5,3,2]
 => ? = 62
[4,3,3,3,3]
 => [5,5,5,1]
 => ? = 64
[3,3,3,3,2,1,1]
 => [7,5,4]
 => ? = 66
[3,3,2,2,2,2,2]
 => [7,7,2]
 => ? = 70
[5,5,5,1,1]
 => [5,3,3,3,3]
 => ? = 69
[5,4,4,4]
 => [4,4,4,4,1]
 => ? = 69
[4,4,3,3,3]
 => [5,5,5,2]
 => ? = 69
[4,4,3,3,2,1]
 => [6,5,4,2]
 => ? = 68
[4,4,3,2,2,2]
 => [6,6,3,2]
 => ? = 69
[3,3,3,3,3,2]
 => [6,6,5]
 => ? = 73
[3,3,3,3,2,2,1]
 => [7,6,4]
 => ? = 73
[3,3,3,2,2,2,2]
 => [7,7,3]
 => ? = 75
[5,5,4,3,1]
 => [5,4,4,3,2]
 => ? = 73
[3,3,3,3,3,2,1]
 => [7,6,5]
 => ? = 80
[4,4,4,3,2,2]
 => [6,6,4,3]
 => ? = 81
[4,4,4,4,3,1]
 => [6,5,5,4]
 => ? = 88
[4,4,4,4,2,1]
 => [6,5,4,4]
 => ? = 81
[3,3,3,3,3,3,3]
 => [7,7,7]
 => ? = 105
[3,3,3,3,3,2,2]
 => [7,7,5]
 => ? = 88
[4,4,4,3,3,3]
 => [6,6,6,3]
 => ? = 96
[4,4,4,4,4,1]
 => [6,5,5,5]
 => ? = 96
[5,5,5,5,1]
 => [5,4,4,4,4]
 => ? = 95
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.
Matching statistic: St000347
(load all 4 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 79% ●values known / values provided: 83%●distinct values known / distinct values provided: 79%
Values
[1]
 => 10 => 1
[2]
 => 100 => 3
[1,1]
 => 110 => 3
[3]
 => 1000 => 6
[2,1]
 => 1010 => 5
[1,1,1]
 => 1110 => 6
[4]
 => 10000 => 10
[3,1]
 => 10010 => 8
[2,2]
 => 1100 => 8
[2,1,1]
 => 10110 => 8
[1,1,1,1]
 => 11110 => 10
[5]
 => 100000 => 15
[4,1]
 => 100010 => 12
[3,2]
 => 10100 => 11
[3,1,1]
 => 100110 => 11
[2,2,1]
 => 11010 => 11
[2,1,1,1]
 => 101110 => 12
[1,1,1,1,1]
 => 111110 => 15
[6]
 => 1000000 => 21
[5,1]
 => 1000010 => 17
[4,2]
 => 100100 => 15
[4,1,1]
 => 1000110 => 15
[3,3]
 => 11000 => 15
[3,2,1]
 => 101010 => 14
[3,1,1,1]
 => 1001110 => 15
[2,2,2]
 => 11100 => 15
[2,2,1,1]
 => 110110 => 15
[2,1,1,1,1]
 => 1011110 => 17
[1,1,1,1,1,1]
 => 1111110 => 21
[7]
 => 10000000 => 28
[6,1]
 => 10000010 => 23
[5,2]
 => 1000100 => 20
[5,1,1]
 => 10000110 => 20
[4,3]
 => 101000 => 19
[4,2,1]
 => 1001010 => 18
[4,1,1,1]
 => 10001110 => 19
[3,3,1]
 => 110010 => 18
[3,2,2]
 => 101100 => 18
[3,2,1,1]
 => 1010110 => 18
[3,1,1,1,1]
 => 10011110 => 20
[2,2,2,1]
 => 111010 => 19
[2,2,1,1,1]
 => 1101110 => 20
[2,1,1,1,1,1]
 => 10111110 => 23
[1,1,1,1,1,1,1]
 => 11111110 => 28
[8]
 => 100000000 => 36
[7,1]
 => 100000010 => 30
[6,2]
 => 10000100 => 26
[6,1,1]
 => 100000110 => 26
[5,3]
 => 1001000 => 24
[5,2,1]
 => 10001010 => 23
[1,1,1,1,1,1,1,1,1]
 => 1111111110 => ? = 45
[2,2,1,1,1,1,1,1]
 => 1101111110 => ? = 41
[2,2,2,1,1,1,1,1]
 => 1110111110 => ? = 45
[3,3,2,1,1,1,1]
 => 1101011110 => ? = 44
[2,2,2,2,1,1,1,1]
 => 1111011110 => ? = 50
[3,3,3,1,1,1,1]
 => 1110011110 => ? = 49
[3,3,2,2,1,1,1]
 => 1101101110 => ? = 49
[2,2,2,2,2,1,1,1]
 => 1111101110 => ? = 56
[4,4,3,1,1,1]
 => 1101001110 => ? = 51
[3,3,3,2,1,1,1]
 => 1110101110 => ? = 54
[3,3,2,2,2,1,1]
 => 1101110110 => ? = 55
[2,2,2,2,2,2,1,1]
 => 1111110110 => ? = 63
[4,4,4,1,1,1]
 => 1110001110 => ? = 57
[4,4,3,2,1,1]
 => 1101010110 => ? = 56
[3,3,3,3,1,1,1]
 => 1111001110 => ? = 60
[3,3,3,2,2,1,1]
 => 1110110110 => ? = 60
[3,3,2,2,2,2,1]
 => 1101111010 => ? = 62
[2,2,2,2,2,2,2,1]
 => 1111111010 => ? = 71
[4,4,4,2,1,1]
 => 1110010110 => ? = 62
[4,4,3,3,1,1]
 => 1101100110 => ? = 62
[4,4,3,2,2,1]
 => 1101011010 => ? = 62
[3,3,3,3,2,1,1]
 => 1111010110 => ? = 66
[3,3,3,2,2,2,1]
 => 1110111010 => ? = 67
[3,3,2,2,2,2,2]
 => 1101111100 => ? = 70
[2,2,2,2,2,2,2,2]
 => 1111111100 => ? = 80
[5,5,5,1,1]
 => 1110000110 => ? = 69
[4,4,4,3,1,1]
 => 1110100110 => ? = 68
[4,4,4,2,2,1]
 => 1110011010 => ? = 68
[4,4,3,3,2,1]
 => 1101101010 => ? = 68
[4,4,3,2,2,2]
 => 1101011100 => ? = 69
[3,3,3,3,2,2,1]
 => 1111011010 => ? = 73
[3,3,3,2,2,2,2]
 => 1110111100 => ? = 75
[4,4,4,3,2,1]
 => 1110101010 => ? = 74
[5,5,4,3,2]
 => 1101010100 => ? = 79
[5,5,4,3,1]
 => 1101010010 => ? = 73
[3,3,3,3,3,2,1]
 => 1111101010 => ? = 80
[4,4,4,3,2,2]
 => 1110101100 => ? = 81
[4,4,4,4,3,1]
 => 1111010010 => ? = 88
[4,4,4,4,2,1]
 => 1111001010 => ? = 81
[3,3,3,3,3,3,1]
 => 1111110010 => ? = 88
[3,3,3,3,3,3,3]
 => 1111111000 => ? = 105
[3,3,3,3,3,2,2]
 => 1111101100 => ? = 88
[4,4,4,3,3,3]
 => 1110111000 => ? = 96
[4,4,4,4,4,4]
 => 1111110000 => ? = 120
[6,6,6,6]
 => 1111000000 => ? = 120
[7,7,7]
 => 1110000000 => ? = 105
[5,5,5,5,5]
 => 1111100000 => ? = 125
[4,4,4,4,4,1]
 => 1111100010 => ? = 96
[5,5,5,5,1]
 => 1111000010 => ? = 95
[6,6,6,1]
 => 1110000010 => ? = 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 $\pi$.