Identifier
Values
0 => [2] => 0
1 => [1,1] => 1
00 => [3] => 0
01 => [2,1] => 2
10 => [1,2] => 1
11 => [1,1,1] => 3
000 => [4] => 0
001 => [3,1] => 3
010 => [2,2] => 2
011 => [2,1,1] => 5
100 => [1,3] => 1
101 => [1,2,1] => 4
110 => [1,1,2] => 3
111 => [1,1,1,1] => 6
0000 => [5] => 0
0001 => [4,1] => 4
0010 => [3,2] => 3
0011 => [3,1,1] => 7
0100 => [2,3] => 2
0101 => [2,2,1] => 6
0110 => [2,1,2] => 5
0111 => [2,1,1,1] => 9
1000 => [1,4] => 1
1001 => [1,3,1] => 5
1010 => [1,2,2] => 4
1011 => [1,2,1,1] => 8
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 7
1110 => [1,1,1,2] => 6
1111 => [1,1,1,1,1] => 10
00000 => [6] => 0
00001 => [5,1] => 5
00010 => [4,2] => 4
00011 => [4,1,1] => 9
00100 => [3,3] => 3
00101 => [3,2,1] => 8
00110 => [3,1,2] => 7
00111 => [3,1,1,1] => 12
01000 => [2,4] => 2
01001 => [2,3,1] => 7
01010 => [2,2,2] => 6
01011 => [2,2,1,1] => 11
01100 => [2,1,3] => 5
01101 => [2,1,2,1] => 10
01110 => [2,1,1,2] => 9
01111 => [2,1,1,1,1] => 14
10000 => [1,5] => 1
10001 => [1,4,1] => 6
10010 => [1,3,2] => 5
10011 => [1,3,1,1] => 10
10100 => [1,2,3] => 4
10101 => [1,2,2,1] => 9
10110 => [1,2,1,2] => 8
10111 => [1,2,1,1,1] => 13
11000 => [1,1,4] => 3
11001 => [1,1,3,1] => 8
11010 => [1,1,2,2] => 7
11011 => [1,1,2,1,1] => 12
11100 => [1,1,1,3] => 6
11101 => [1,1,1,2,1] => 11
11110 => [1,1,1,1,2] => 10
11111 => [1,1,1,1,1,1] => 15
000000 => [7] => 0
000001 => [6,1] => 6
000010 => [5,2] => 5
000011 => [5,1,1] => 11
000100 => [4,3] => 4
000101 => [4,2,1] => 10
000110 => [4,1,2] => 9
000111 => [4,1,1,1] => 15
001000 => [3,4] => 3
001001 => [3,3,1] => 9
001010 => [3,2,2] => 8
001011 => [3,2,1,1] => 14
001100 => [3,1,3] => 7
001101 => [3,1,2,1] => 13
001110 => [3,1,1,2] => 12
001111 => [3,1,1,1,1] => 18
010000 => [2,5] => 2
010001 => [2,4,1] => 8
010010 => [2,3,2] => 7
010011 => [2,3,1,1] => 13
010100 => [2,2,3] => 6
010101 => [2,2,2,1] => 12
010110 => [2,2,1,2] => 11
010111 => [2,2,1,1,1] => 17
011000 => [2,1,4] => 5
011001 => [2,1,3,1] => 11
011010 => [2,1,2,2] => 10
011011 => [2,1,2,1,1] => 16
011100 => [2,1,1,3] => 9
011101 => [2,1,1,2,1] => 15
011110 => [2,1,1,1,2] => 14
011111 => [2,1,1,1,1,1] => 20
100000 => [1,6] => 1
100001 => [1,5,1] => 7
100010 => [1,4,2] => 6
100011 => [1,4,1,1] => 12
100100 => [1,3,3] => 5
100101 => [1,3,2,1] => 11
100110 => [1,3,1,2] => 10
>>> Load all 405 entries. <<<
100111 => [1,3,1,1,1] => 16
101000 => [1,2,4] => 4
101001 => [1,2,3,1] => 10
101010 => [1,2,2,2] => 9
101011 => [1,2,2,1,1] => 15
101100 => [1,2,1,3] => 8
101101 => [1,2,1,2,1] => 14
101110 => [1,2,1,1,2] => 13
101111 => [1,2,1,1,1,1] => 19
110000 => [1,1,5] => 3
110001 => [1,1,4,1] => 9
110010 => [1,1,3,2] => 8
110011 => [1,1,3,1,1] => 14
110100 => [1,1,2,3] => 7
110101 => [1,1,2,2,1] => 13
110110 => [1,1,2,1,2] => 12
110111 => [1,1,2,1,1,1] => 18
111000 => [1,1,1,4] => 6
111001 => [1,1,1,3,1] => 12
111010 => [1,1,1,2,2] => 11
111011 => [1,1,1,2,1,1] => 17
111100 => [1,1,1,1,3] => 10
111101 => [1,1,1,1,2,1] => 16
111110 => [1,1,1,1,1,2] => 15
111111 => [1,1,1,1,1,1,1] => 21
0000000 => [8] => 0
0000001 => [7,1] => 7
0000010 => [6,2] => 6
0000011 => [6,1,1] => 13
0000100 => [5,3] => 5
0000101 => [5,2,1] => 12
0000110 => [5,1,2] => 11
0000111 => [5,1,1,1] => 18
0001000 => [4,4] => 4
0001001 => [4,3,1] => 11
0001010 => [4,2,2] => 10
0001011 => [4,2,1,1] => 17
0001100 => [4,1,3] => 9
0001101 => [4,1,2,1] => 16
0001110 => [4,1,1,2] => 15
0001111 => [4,1,1,1,1] => 22
0010000 => [3,5] => 3
0010001 => [3,4,1] => 10
0010010 => [3,3,2] => 9
0010011 => [3,3,1,1] => 16
0010100 => [3,2,3] => 8
0010101 => [3,2,2,1] => 15
0010110 => [3,2,1,2] => 14
0010111 => [3,2,1,1,1] => 21
0011000 => [3,1,4] => 7
0011001 => [3,1,3,1] => 14
0011010 => [3,1,2,2] => 13
0011011 => [3,1,2,1,1] => 20
0011100 => [3,1,1,3] => 12
0011101 => [3,1,1,2,1] => 19
0011110 => [3,1,1,1,2] => 18
0011111 => [3,1,1,1,1,1] => 25
0100000 => [2,6] => 2
0100001 => [2,5,1] => 9
0100010 => [2,4,2] => 8
0100011 => [2,4,1,1] => 15
0100100 => [2,3,3] => 7
0100101 => [2,3,2,1] => 14
0100110 => [2,3,1,2] => 13
0100111 => [2,3,1,1,1] => 20
0101000 => [2,2,4] => 6
0101001 => [2,2,3,1] => 13
0101010 => [2,2,2,2] => 12
0101011 => [2,2,2,1,1] => 19
0101100 => [2,2,1,3] => 11
0101101 => [2,2,1,2,1] => 18
0101110 => [2,2,1,1,2] => 17
0101111 => [2,2,1,1,1,1] => 24
0110000 => [2,1,5] => 5
0110001 => [2,1,4,1] => 12
0110010 => [2,1,3,2] => 11
0110011 => [2,1,3,1,1] => 18
0110100 => [2,1,2,3] => 10
0110101 => [2,1,2,2,1] => 17
0110110 => [2,1,2,1,2] => 16
0110111 => [2,1,2,1,1,1] => 23
0111000 => [2,1,1,4] => 9
0111001 => [2,1,1,3,1] => 16
0111010 => [2,1,1,2,2] => 15
0111011 => [2,1,1,2,1,1] => 22
0111100 => [2,1,1,1,3] => 14
0111101 => [2,1,1,1,2,1] => 21
0111110 => [2,1,1,1,1,2] => 20
0111111 => [2,1,1,1,1,1,1] => 27
1000000 => [1,7] => 1
1000001 => [1,6,1] => 8
1000010 => [1,5,2] => 7
1000011 => [1,5,1,1] => 14
1000100 => [1,4,3] => 6
1000101 => [1,4,2,1] => 13
1000110 => [1,4,1,2] => 12
1000111 => [1,4,1,1,1] => 19
1001000 => [1,3,4] => 5
1001001 => [1,3,3,1] => 12
1001010 => [1,3,2,2] => 11
1001011 => [1,3,2,1,1] => 18
1001100 => [1,3,1,3] => 10
1001101 => [1,3,1,2,1] => 17
1001110 => [1,3,1,1,2] => 16
1001111 => [1,3,1,1,1,1] => 23
1010000 => [1,2,5] => 4
1010001 => [1,2,4,1] => 11
1010010 => [1,2,3,2] => 10
1010011 => [1,2,3,1,1] => 17
1010100 => [1,2,2,3] => 9
1010101 => [1,2,2,2,1] => 16
1010110 => [1,2,2,1,2] => 15
1010111 => [1,2,2,1,1,1] => 22
1011000 => [1,2,1,4] => 8
1011001 => [1,2,1,3,1] => 15
1011010 => [1,2,1,2,2] => 14
1011011 => [1,2,1,2,1,1] => 21
1011100 => [1,2,1,1,3] => 13
1011101 => [1,2,1,1,2,1] => 20
1011110 => [1,2,1,1,1,2] => 19
1011111 => [1,2,1,1,1,1,1] => 26
1100000 => [1,1,6] => 3
1100001 => [1,1,5,1] => 10
1100010 => [1,1,4,2] => 9
1100011 => [1,1,4,1,1] => 16
1100100 => [1,1,3,3] => 8
1100101 => [1,1,3,2,1] => 15
1100110 => [1,1,3,1,2] => 14
1100111 => [1,1,3,1,1,1] => 21
1101000 => [1,1,2,4] => 7
1101001 => [1,1,2,3,1] => 14
1101010 => [1,1,2,2,2] => 13
1101011 => [1,1,2,2,1,1] => 20
1101100 => [1,1,2,1,3] => 12
1101101 => [1,1,2,1,2,1] => 19
1101110 => [1,1,2,1,1,2] => 18
1101111 => [1,1,2,1,1,1,1] => 25
1110000 => [1,1,1,5] => 6
1110001 => [1,1,1,4,1] => 13
1110010 => [1,1,1,3,2] => 12
1110011 => [1,1,1,3,1,1] => 19
1110100 => [1,1,1,2,3] => 11
1110101 => [1,1,1,2,2,1] => 18
1110110 => [1,1,1,2,1,2] => 17
1110111 => [1,1,1,2,1,1,1] => 24
1111000 => [1,1,1,1,4] => 10
1111001 => [1,1,1,1,3,1] => 17
1111010 => [1,1,1,1,2,2] => 16
1111011 => [1,1,1,1,2,1,1] => 23
1111100 => [1,1,1,1,1,3] => 15
1111101 => [1,1,1,1,1,2,1] => 22
1111110 => [1,1,1,1,1,1,2] => 21
1111111 => [1,1,1,1,1,1,1,1] => 28
00000001 => [8,1] => 8
00000011 => [7,1,1] => 15
00000101 => [6,2,1] => 14
00110001 => [3,1,4,1] => 15
01010001 => [2,2,4,1] => 14
01100001 => [2,1,5,1] => 13
10000000 => [1,8] => 1
10000001 => [1,7,1] => 9
10000010 => [1,6,2] => 8
10000011 => [1,6,1,1] => 16
10000101 => [1,5,2,1] => 15
10000110 => [1,5,1,2] => 14
10001001 => [1,4,3,1] => 14
10001010 => [1,4,2,2] => 13
10001100 => [1,4,1,3] => 12
10001111 => [1,4,1,1,1,1] => 27
10010001 => [1,3,4,1] => 13
10010111 => [1,3,2,1,1,1] => 26
10011011 => [1,3,1,2,1,1] => 25
10011101 => [1,3,1,1,2,1] => 24
10100001 => [1,2,5,1] => 12
10101101 => [1,2,2,1,2,1] => 23
10110101 => [1,2,1,2,2,1] => 22
10111001 => [1,2,1,1,3,1] => 21
11000001 => [1,1,6,1] => 11
11011001 => [1,1,2,1,3,1] => 20
11101001 => [1,1,1,2,3,1] => 19
11110001 => [1,1,1,1,4,1] => 18
000000001 => [9,1] => 9
000000011 => [8,1,1] => 17
000010000 => [5,5] => 5
000100011 => [4,4,1,1] => 21
000100110 => [4,3,1,2] => 19
000101100 => [4,2,1,3] => 17
000111000 => [4,1,1,4] => 15
001001010 => [3,3,2,2] => 17
001001111 => [3,3,1,1,1,1] => 33
001010100 => [3,2,2,3] => 15
001011011 => [3,2,1,2,1,1] => 31
001011110 => [3,2,1,1,1,2] => 29
001101000 => [3,1,2,4] => 13
001110011 => [3,1,1,3,1,1] => 29
001110110 => [3,1,1,2,1,2] => 27
001111100 => [3,1,1,1,1,3] => 25
010100100 => [2,2,3,3] => 13
010101011 => [2,2,2,2,1,1] => 29
010101110 => [2,2,2,1,1,2] => 27
010111010 => [2,2,1,1,2,2] => 25
010111111 => [2,2,1,1,1,1,1,1] => 41
011000001 => [2,1,6,1] => 14
011001000 => [2,1,3,4] => 11
011010011 => [2,1,2,3,1,1] => 27
011010110 => [2,1,2,2,1,2] => 25
011011100 => [2,1,2,1,1,3] => 23
011101010 => [2,1,1,2,2,2] => 23
011101111 => [2,1,1,2,1,1,1,1] => 39
011110100 => [2,1,1,1,2,3] => 21
011111011 => [2,1,1,1,1,2,1,1] => 37
011111110 => [2,1,1,1,1,1,1,2] => 35
100000000 => [1,9] => 1
100000001 => [1,8,1] => 10
100000011 => [1,7,1,1] => 18
100000101 => [1,6,2,1] => 17
100000110 => [1,6,1,2] => 16
100001001 => [1,5,3,1] => 16
100001111 => [1,5,1,1,1,1] => 31
100010001 => [1,4,4,1] => 15
100011101 => [1,4,1,1,2,1] => 28
100100001 => [1,3,5,1] => 14
100111001 => [1,3,1,1,3,1] => 25
101000001 => [1,2,6,1] => 13
101110001 => [1,2,1,1,4,1] => 22
110000001 => [1,1,7,1] => 12
110001000 => [1,1,4,4] => 9
110010011 => [1,1,3,3,1,1] => 25
110010110 => [1,1,3,2,1,2] => 23
110011100 => [1,1,3,1,1,3] => 21
110101010 => [1,1,2,2,2,2] => 21
110101111 => [1,1,2,2,1,1,1,1] => 37
110110100 => [1,1,2,1,2,3] => 19
110111011 => [1,1,2,1,1,2,1,1] => 35
110111110 => [1,1,2,1,1,1,1,2] => 33
111100001 => [1,1,1,1,5,1] => 19
111100100 => [1,1,1,1,3,3] => 17
111101011 => [1,1,1,1,2,2,1,1] => 33
111101110 => [1,1,1,1,2,1,1,2] => 31
111111010 => [1,1,1,1,1,1,2,2] => 29
111111111 => [1,1,1,1,1,1,1,1,1,1] => 45
1100000001 => [1,1,8,1] => 13
1010000001 => [1,2,7,1] => 14
1000001001 => [1,6,3,1] => 18
1000000101 => [1,7,2,1] => 19
1000000011 => [1,8,1,1] => 20
1000000000 => [1,10] => 1
11011111110 => [1,1,2,1,1,1,1,1,1,2] => 52
11110111110 => [1,1,1,1,2,1,1,1,1,2] => 50
11111101110 => [1,1,1,1,1,1,2,1,1,2] => 48
11010101110 => [1,1,2,2,2,1,1,2] => 40
11111111010 => [1,1,1,1,1,1,1,1,2,2] => 46
=> [1] => 0
0000000001 => [10,1] => 10
11010100100 => [1,1,2,2,3,3] => 22
11001010100 => [1,1,3,2,2,3] => 24
00101010011 => [3,2,2,3,1,1] => 36
00100101011 => [3,3,2,2,1,1] => 38
01111111110 => [2,1,1,1,1,1,1,1,1,2] => 54
01010101010 => [2,2,2,2,2,2] => 30
01010001000 => [2,2,4,4] => 14
00100100100 => [3,3,3,3] => 18
00101001000 => [3,2,3,4] => 16
00010001010 => [4,4,2,2] => 22
00010010100 => [4,3,2,3] => 20
00010101000 => [4,2,2,4] => 18
00000100000 => [6,6] => 6
01011111111 => [2,2,1,1,1,1,1,1,1,1] => 62
01111111011 => [2,1,1,1,1,1,1,2,1,1] => 56
11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => 66
01010111011 => [2,2,2,1,1,2,1,1] => 48
01011101110 => [2,2,1,1,2,1,1,2] => 44
01011101011 => [2,2,1,1,2,2,1,1] => 46
01101110110 => [2,1,2,1,1,2,1,2] => 42
01101011011 => [2,1,2,2,1,2,1,1] => 46
01110111010 => [2,1,1,2,1,1,2,2] => 40
01110101110 => [2,1,1,2,2,1,1,2] => 42
01110101011 => [2,1,1,2,2,2,1,1] => 44
01111101111 => [2,1,1,1,1,2,1,1,1,1] => 58
11011101010 => [1,1,2,1,1,2,2,2] => 36
11011111011 => [1,1,2,1,1,1,1,2,1,1] => 54
11011010110 => [1,1,2,1,2,2,1,2] => 38
11010111010 => [1,1,2,2,1,1,2,2] => 38
11010101011 => [1,1,2,2,2,2,1,1] => 42
11011101111 => [1,1,2,1,1,2,1,1,1,1] => 56
11110111011 => [1,1,1,1,2,1,1,2,1,1] => 52
11110101111 => [1,1,1,1,2,2,1,1,1,1] => 54
00101111110 => [3,2,1,1,1,1,1,2] => 48
00111011110 => [3,1,1,2,1,1,1,2] => 46
00111110110 => [3,1,1,1,1,2,1,2] => 44
00111111100 => [3,1,1,1,1,1,1,3] => 42
01101010100 => [2,1,2,2,2,3] => 26
01101001010 => [2,1,2,3,2,2] => 28
01010101111 => [2,2,2,2,1,1,1,1] => 50
11011110100 => [1,1,2,1,1,1,2,3] => 34
01010110100 => [2,2,2,1,2,3] => 28
01110111111 => [2,1,1,2,1,1,1,1,1,1] => 60
11110001000 => [1,1,1,1,4,4] => 18
11000111000 => [1,1,4,1,1,4] => 24
11000100011 => [1,1,4,4,1,1] => 30
00011111000 => [4,1,1,1,1,4] => 30
00011100011 => [4,1,1,4,1,1] => 36
00010001111 => [4,4,1,1,1,1] => 42
01011111010 => [2,2,1,1,1,1,2,2] => 42
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Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.