Identifier
-
Mp00093:
Dyck paths
—to binary word⟶
Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000769: Integer compositions ⟶ ℤ
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..
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
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.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
See Mp00096Foata bijection.
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