Identifier
Values
=>
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101011110=>100111110=>9 101011111=>011011111=>3 101100000=>001010010=>16 101100001=>010100101=>13 101100010=>010100110=>14 101100011=>101001011=>10 101100100=>010110010=>15 101100101=>101001101=>11 101100110=>101001110=>12 101100111=>110010111=>7 101101000=>011010010=>16 101101001=>101100101=>12 101101010=>101100110=>13 101101011=>110011011=>8 101101100=>101110010=>14 101101101=>110011101=>9 101101110=>110011110=>10 101101111=>011101111=>4 101110000=>101010010=>17 101110001=>110100101=>13 101110010=>110100110=>14 101110011=>111001011=>9 101110100=>110110010=>15 101110101=>111001101=>10 101110110=>111001110=>11 101110111=>011110111=>5 101111000=>111010010=>16 101111001=>111100101=>11 101111010=>111100110=>12 101111011=>011111011=>6 101111100=>111110010=>13 101111101=>011111101=>7 101111110=>011111110=>8 101111111=>101111111=>1 110000000=>000001010=>14 110000001=>000010101=>12 110000010=>000010110=>13 110000011=>000101011=>10 110000100=>000011010=>14 110000101=>000101101=>11 110000110=>000101110=>12 110000111=>001010111=>8 110001000=>100001010=>15 110001001=>000110101=>12 110001010=>000110110=>13 110001011=>001011011=>9 110001100=>000111010=>14 110001101=>001011101=>10 110001110=>001011110=>11 110001111=>010101111=>6 110010000=>010001010=>16 110010001=>100010101=>13 110010010=>100010110=>14 110010011=>001101011=>10 110010100=>100011010=>15 110010101=>001101101=>11 110010110=>001101110=>12 110010111=>010110111=>7 110011000=>110001010=>16 110011001=>001110101=>12 110011010=>001110110=>13 110011011=>010111011=>8 110011100=>001111010=>14 110011101=>010111101=>9 110011110=>010111110=>10 110011111=>101011111=>4 110100000=>001001010=>17 110100001=>010010101=>14 110100010=>010010110=>15 110100011=>100101011=>11 110100100=>010011010=>16 110100101=>100101101=>12 110100110=>100101110=>13 110100111=>011010111=>8 110101000=>011001010=>17 110101001=>100110101=>13 110101010=>100110110=>14 110101011=>011011011=>9 110101100=>100111010=>15 110101101=>011011101=>10 110101110=>011011110=>11 110101111=>101101111=>5 110110000=>101001010=>18 110110001=>110010101=>14 110110010=>110010110=>15 110110011=>011101011=>10 110110100=>110011010=>16 110110101=>011101101=>11 110110110=>011101110=>12 110110111=>101110111=>6 110111000=>111001010=>17 110111001=>011110101=>12 110111010=>011110110=>13 110111011=>101111011=>7 110111100=>011111010=>14 110111101=>101111101=>8 110111110=>101111110=>9 110111111=>110111111=>2 111000000=>000101010=>18 111000001=>001010101=>15 111000010=>001010110=>16 111000011=>010101011=>12 111000100=>001011010=>17 111000101=>010101101=>13 111000110=>010101110=>14 111000111=>101010111=>9 111001000=>001101010=>18 111001001=>010110101=>14 111001010=>010110110=>15 111001011=>101011011=>10 111001100=>010111010=>16 111001101=>101011101=>11 111001110=>101011110=>12 111001111=>110101111=>6 111010000=>100101010=>19 111010001=>011010101=>15 111010010=>011010110=>16 111010011=>101101011=>11 111010100=>011011010=>17 111010101=>101101101=>12 111010110=>101101110=>13 111010111=>110110111=>7 111011000=>011101010=>18 111011001=>101110101=>13 111011010=>101110110=>14 111011011=>110111011=>8 111011100=>101111010=>15 111011101=>110111101=>9 111011110=>110111110=>10 111011111=>111011111=>3 111100000=>010101010=>20 111100001=>101010101=>16 111100010=>101010110=>17 111100011=>110101011=>12 111100100=>101011010=>18 111100101=>110101101=>13 111100110=>110101110=>14 111100111=>111010111=>8 111101000=>101101010=>19 111101001=>110110101=>14 111101010=>110110110=>15 111101011=>111011011=>9 111101100=>110111010=>16 111101101=>111011101=>10 111101110=>111011110=>11 111101111=>111101111=>4 111110000=>110101010=>20 111110001=>111010101=>15 111110010=>111010110=>16 111110011=>111101011=>10 111110100=>111011010=>17 111110101=>111101101=>11 111110110=>111101110=>12 111110111=>111110111=>5 111111000=>111101010=>18 111111001=>111110101=>12 111111010=>111110110=>13 111111011=>111111011=>6 111111100=>111111010=>14 111111101=>111111101=>7 111111110=>111111110=>8 111111111=>111111111=>0 1111111110=>1111111110=>9 1101111110=>1011111110=>10 1010111110=>1001111110=>10 1001011110=>1000111110=>10 1000111110=>0001111110=>9 1000101110=>1000011110=>10 1000011110=>0000111110=>9 1000010110=>1000001110=>10 1000001110=>0000011110=>9 1000001010=>1000000110=>10 1000000110=>0000001110=>9 1000000100=>1000000010=>10 1000000010=>0000000110=>9 1000000000=>0000000010=>9 10000000000=>00000000010=>10 11111111110=>11111111110=>10 0111111111=>0111111111=>0 0111100001=>1010101001=>16 0111000001=>0010101001=>15 0110000001=>0000101001=>12 0100000001=>0000001001=>7 0000000001=>0000000001=>0 01111111111=>01111111111=>0 0111010001=>0110101001=>15 01111000001=>01010101001=>20 0101000001=>0001001001=>11 0010000001=>0000010001=>6 00000000001=>00000000001=>0
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Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.