Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
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.
Mp00095: Integer partitions to binary wordBinary words
St000347: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
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
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 $\pi$.
Mp00095: Integer partitions to binary wordBinary words
Mp00105: Binary words complementBinary 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 $w$.