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 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.