Identifier
Values
[1,0] => 10 => 10 => [1,2] => 0
[1,0,1,0] => 1010 => 0110 => [2,1,2] => 1
[1,1,0,0] => 1100 => 1010 => [1,2,2] => 0
[1,0,1,0,1,0] => 101010 => 100110 => [1,3,1,2] => 2
[1,0,1,1,0,0] => 101100 => 110010 => [1,1,3,2] => 3
[1,1,0,0,1,0] => 110010 => 010110 => [2,2,1,2] => 2
[1,1,0,1,0,0] => 110100 => 011010 => [2,1,2,2] => 1
[1,1,1,0,0,0] => 111000 => 101010 => [1,2,2,2] => 0
[1,0,1,0,1,0,1,0] => 10101010 => 01100110 => [2,1,3,1,2] => 4
[1,0,1,0,1,1,0,0] => 10101100 => 01110010 => [2,1,1,3,2] => 5
[1,0,1,1,0,0,1,0] => 10110010 => 10100110 => [1,2,3,1,2] => 3
[1,0,1,1,0,1,0,0] => 10110100 => 10110010 => [1,2,1,3,2] => 6
[1,0,1,1,1,0,0,0] => 10111000 => 11010010 => [1,1,2,3,2] => 4
[1,1,0,0,1,0,1,0] => 11001010 => 00110110 => [3,1,2,1,2] => 4
[1,1,0,0,1,1,0,0] => 11001100 => 00111010 => [3,1,1,2,2] => 1
[1,1,0,1,0,0,1,0] => 11010010 => 10010110 => [1,3,2,1,2] => 5
[1,1,0,1,0,1,0,0] => 11010100 => 10011010 => [1,3,1,2,2] => 2
[1,1,0,1,1,0,0,0] => 11011000 => 11001010 => [1,1,3,2,2] => 3
[1,1,1,0,0,0,1,0] => 11100010 => 01010110 => [2,2,2,1,2] => 3
[1,1,1,0,0,1,0,0] => 11100100 => 01011010 => [2,2,1,2,2] => 2
[1,1,1,0,1,0,0,0] => 11101000 => 01101010 => [2,1,2,2,2] => 1
[1,1,1,1,0,0,0,0] => 11110000 => 10101010 => [1,2,2,2,2] => 0
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Description
The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see St000008The major index of the composition..
Map
to binary word
Description
Return the Dyck word as binary word.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
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.