Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000869
(load all 3 compositions to match this statistic)
Mapping
St000869: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 3
[1,1]
 => 3
[3]
 => 6
[2,1]
 => 5
[1,1,1]
 => 6
[4]
 => 10
[3,1]
 => 8
[2,2]
 => 8
[2,1,1]
 => 8
[1,1,1,1]
 => 10
[5]
 => 15
[4,1]
 => 12
[3,2]
 => 11
[3,1,1]
 => 11
[2,2,1]
 => 11
[2,1,1,1]
 => 12
[1,1,1,1,1]
 => 15
[6]
 => 21
[5,1]
 => 17
[4,2]
 => 15
[4,1,1]
 => 15
[3,3]
 => 15
[3,2,1]
 => 14
[3,1,1,1]
 => 15
[2,2,2]
 => 15
[2,2,1,1]
 => 15
[2,1,1,1,1]
 => 17
[1,1,1,1,1,1]
 => 21
[7]
 => 28
[6,1]
 => 23
[5,2]
 => 20
[5,1,1]
 => 20
[4,3]
 => 19
[4,2,1]
 => 18
[4,1,1,1]
 => 19
[3,3,1]
 => 18
[3,2,2]
 => 18
[3,2,1,1]
 => 18
[3,1,1,1,1]
 => 20
[2,2,2,1]
 => 19
[2,2,1,1,1]
 => 20
[2,1,1,1,1,1]
 => 23
[1,1,1,1,1,1,1]
 => 28
[8]
 => 36
[7,1]
 => 30
[6,2]
 => 26
[6,1,1]
 => 26
[5,3]
 => 24
[5,2,1]
 => 23
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: St000348
(load all 2 compositions to match this statistic)
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000348: Binary words ⟶ ℤResult quality: 28% ●values known / values provided: 34%●distinct values known / distinct values provided: 28%
Values
[1]
 => 10 => 01 => 1
[2]
 => 100 => 001 => 3
[1,1]
 => 110 => 011 => 3
[3]
 => 1000 => 0001 => 6
[2,1]
 => 1010 => 0101 => 5
[1,1,1]
 => 1110 => 0111 => 6
[4]
 => 10000 => 00001 => 10
[3,1]
 => 10010 => 01001 => 8
[2,2]
 => 1100 => 0011 => 8
[2,1,1]
 => 10110 => 01101 => 8
[1,1,1,1]
 => 11110 => 01111 => 10
[5]
 => 100000 => 000001 => 15
[4,1]
 => 100010 => 010001 => 12
[3,2]
 => 10100 => 00101 => 11
[3,1,1]
 => 100110 => 011001 => 11
[2,2,1]
 => 11010 => 01011 => 11
[2,1,1,1]
 => 101110 => 011101 => 12
[1,1,1,1,1]
 => 111110 => 011111 => 15
[6]
 => 1000000 => 0000001 => 21
[5,1]
 => 1000010 => 0100001 => 17
[4,2]
 => 100100 => 001001 => 15
[4,1,1]
 => 1000110 => 0110001 => 15
[3,3]
 => 11000 => 00011 => 15
[3,2,1]
 => 101010 => 010101 => 14
[3,1,1,1]
 => 1001110 => 0111001 => 15
[2,2,2]
 => 11100 => 00111 => 15
[2,2,1,1]
 => 110110 => 011011 => 15
[2,1,1,1,1]
 => 1011110 => 0111101 => 17
[1,1,1,1,1,1]
 => 1111110 => 0111111 => 21
[7]
 => 10000000 => 00000001 => 28
[6,1]
 => 10000010 => 01000001 => 23
[5,2]
 => 1000100 => 0010001 => 20
[5,1,1]
 => 10000110 => 01100001 => 20
[4,3]
 => 101000 => 000101 => 19
[4,2,1]
 => 1001010 => 0101001 => 18
[4,1,1,1]
 => 10001110 => 01110001 => 19
[3,3,1]
 => 110010 => 010011 => 18
[3,2,2]
 => 101100 => 001101 => 18
[3,2,1,1]
 => 1010110 => 0110101 => 18
[3,1,1,1,1]
 => 10011110 => 01111001 => 20
[2,2,2,1]
 => 111010 => 010111 => 19
[2,2,1,1,1]
 => 1101110 => 0111011 => 20
[2,1,1,1,1,1]
 => 10111110 => 01111101 => 23
[1,1,1,1,1,1,1]
 => 11111110 => 01111111 => 28
[8]
 => 100000000 => 000000001 => 36
[7,1]
 => 100000010 => 010000001 => 30
[6,2]
 => 10000100 => 00100001 => 26
[6,1,1]
 => 100000110 => 011000001 => 26
[5,3]
 => 1001000 => 0001001 => 24
[5,2,1]
 => 10001010 => 01010001 => 23
[8,1]
 => 1000000010 => 0100000001 => ? = 38
[7,1,1]
 => 1000000110 => 0110000001 => ? = 33
[6,1,1,1]
 => 1000001110 => 0111000001 => ? = 30
[5,1,1,1,1]
 => 1000011110 => 0111100001 => ? = 29
[4,1,1,1,1,1]
 => 1000111110 => 0111110001 => ? = 30
[3,1,1,1,1,1,1]
 => 1001111110 => 0111111001 => ? = 33
[2,1,1,1,1,1,1,1]
 => 1011111110 => 0111111101 => ? = 38
[1,1,1,1,1,1,1,1,1]
 => 1111111110 => 0111111111 => ? = 45
[10]
 => 10000000000 => 00000000001 => ? = 55
[9,1]
 => 10000000010 => 01000000001 => ? = 47
[8,1,1]
 => 10000000110 => 01100000001 => ? = 41
[7,2,1]
 => 1000001010 => 0101000001 => ? = 36
[7,1,1,1]
 => 10000001110 => 01110000001 => ? = 37
[6,2,1,1]
 => 1000010110 => 0110100001 => ? = 33
[6,1,1,1,1]
 => 10000011110 => 01111000001 => ? = 35
[5,2,1,1,1]
 => 1000101110 => 0111010001 => ? = 32
[5,1,1,1,1,1]
 => 10000111110 => 01111100001 => ? = 35
[4,2,1,1,1,1]
 => 1001011110 => 0111101001 => ? = 33
[4,1,1,1,1,1,1]
 => 10001111110 => 01111110001 => ? = 37
[3,2,1,1,1,1,1]
 => 1010111110 => 0111110101 => ? = 36
[3,1,1,1,1,1,1,1]
 => 10011111110 => 01111111001 => ? = 41
[2,2,1,1,1,1,1,1]
 => 1101111110 => 0111111011 => ? = 41
[2,1,1,1,1,1,1,1,1]
 => 10111111110 => 01111111101 => ? = 47
[1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => 01111111111 => ? = 55
[11]
 => 100000000000 => 000000000001 => ? = 66
[10,1]
 => 100000000010 => 010000000001 => ? = 57
[9,2]
 => 10000000100 => 00100000001 => ? = 50
[9,1,1]
 => 100000000110 => 011000000001 => ? = 50
[8,2,1]
 => 10000001010 => 01010000001 => ? = 44
[8,1,1,1]
 => 100000001110 => 011100000001 => ? = 45
[7,3,1]
 => 1000010010 => 0100100001 => ? = 40
[7,2,2]
 => 1000001100 => 0011000001 => ? = 40
[7,2,1,1]
 => 10000010110 => 01101000001 => ? = 40
[7,1,1,1,1]
 => 100000011110 => 011110000001 => ? = 42
[6,3,1,1]
 => 1000100110 => 0110010001 => ? = 37
[6,2,2,1]
 => 1000011010 => 0101100001 => ? = 37
[6,2,1,1,1]
 => 10000101110 => 01110100001 => ? = 38
[6,1,1,1,1,1]
 => 100000111110 => 011111000001 => ? = 41
[5,3,1,1,1]
 => 1001001110 => 0111001001 => ? = 36
[5,2,2,1,1]
 => 1000110110 => 0110110001 => ? = 36
[5,2,1,1,1,1]
 => 10001011110 => 01111010001 => ? = 38
[5,1,1,1,1,1,1]
 => 100001111110 => 011111100001 => ? = 42
[4,3,1,1,1,1]
 => 1010011110 => 0111100101 => ? = 37
[4,2,2,1,1,1]
 => 1001101110 => 0111011001 => ? = 37
[4,2,1,1,1,1,1]
 => 10010111110 => 01111101001 => ? = 40
[4,1,1,1,1,1,1,1]
 => 100011111110 => 011111110001 => ? = 45
[3,3,1,1,1,1,1]
 => 1100111110 => 0111110011 => ? = 40
[3,2,2,1,1,1,1]
 => 1011011110 => 0111101101 => ? = 40
[3,2,1,1,1,1,1,1]
 => 10101111110 => 01111110101 => ? = 44
[3,1,1,1,1,1,1,1,1]
 => 100111111110 => 011111111001 => ? = 50
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
Mp00095: Integer partitions —to binary word⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 22% ●values known / values provided: 31%●distinct values known / distinct values provided: 22%
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
[9]
 => 1000000000 => ? = 45
[8,1]
 => 1000000010 => ? = 38
[7,1,1]
 => 1000000110 => ? = 33
[6,1,1,1]
 => 1000001110 => ? = 30
[5,1,1,1,1]
 => 1000011110 => ? = 29
[4,1,1,1,1,1]
 => 1000111110 => ? = 30
[3,1,1,1,1,1,1]
 => 1001111110 => ? = 33
[2,1,1,1,1,1,1,1]
 => 1011111110 => ? = 38
[1,1,1,1,1,1,1,1,1]
 => 1111111110 => ? = 45
[10]
 => 10000000000 => ? = 55
[9,1]
 => 10000000010 => ? = 47
[8,2]
 => 1000000100 => ? = 41
[8,1,1]
 => 10000000110 => ? = 41
[7,2,1]
 => 1000001010 => ? = 36
[7,1,1,1]
 => 10000001110 => ? = 37
[6,2,1,1]
 => 1000010110 => ? = 33
[6,1,1,1,1]
 => 10000011110 => ? = 35
[5,2,1,1,1]
 => 1000101110 => ? = 32
[5,1,1,1,1,1]
 => 10000111110 => ? = 35
[4,2,1,1,1,1]
 => 1001011110 => ? = 33
[4,1,1,1,1,1,1]
 => 10001111110 => ? = 37
[3,2,1,1,1,1,1]
 => 1010111110 => ? = 36
[3,1,1,1,1,1,1,1]
 => 10011111110 => ? = 41
[2,2,1,1,1,1,1,1]
 => 1101111110 => ? = 41
[2,1,1,1,1,1,1,1,1]
 => 10111111110 => ? = 47
[1,1,1,1,1,1,1,1,1,1]
 => 11111111110 => ? = 55
[11]
 => 100000000000 => ? = 66
[10,1]
 => 100000000010 => ? = 57
[9,2]
 => 10000000100 => ? = 50
[9,1,1]
 => 100000000110 => ? = 50
[8,3]
 => 1000001000 => ? = 45
[8,2,1]
 => 10000001010 => ? = 44
[8,1,1,1]
 => 100000001110 => ? = 45
[7,3,1]
 => 1000010010 => ? = 40
[7,2,2]
 => 1000001100 => ? = 40
[7,2,1,1]
 => 10000010110 => ? = 40
[7,1,1,1,1]
 => 100000011110 => ? = 42
[6,3,1,1]
 => 1000100110 => ? = 37
[6,2,2,1]
 => 1000011010 => ? = 37
[6,2,1,1,1]
 => 10000101110 => ? = 38
[6,1,1,1,1,1]
 => 100000111110 => ? = 41
[5,3,1,1,1]
 => 1001001110 => ? = 36
[5,2,2,1,1]
 => 1000110110 => ? = 36
[5,2,1,1,1,1]
 => 10001011110 => ? = 38
[5,1,1,1,1,1,1]
 => 100001111110 => ? = 42
[4,3,1,1,1,1]
 => 1010011110 => ? = 37
[4,2,2,1,1,1]
 => 1001101110 => ? = 37
[4,2,1,1,1,1,1]
 => 10010111110 => ? = 40
[4,1,1,1,1,1,1,1]
 => 100011111110 => ? = 45
[3,3,1,1,1,1,1]
 => 1100111110 => ? = 40
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 $\pi$.