Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
St000008: Integer compositions ⟶ ℤ
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
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.
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.
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.
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