Identifier
Values
[1,0] => [[1],[2]] => 1 => 1 => 1
[1,0,1,0] => [[1,3],[2,4]] => 101 => 101 => 2
[1,1,0,0] => [[1,2],[3,4]] => 010 => 100 => 1
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 10101 => 01101 => 3
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 10010 => 00110 => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 01001 => 01001 => 2
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 01010 => 01100 => 2
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 00100 => 01000 => 1
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 1010101 => 1001101 => 4
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 1010010 => 0100110 => 3
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1001001 => 1000101 => 3
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 1001010 => 1000110 => 3
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 1000100 => 1000010 => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 0100101 => 0011001 => 3
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 0100010 => 0001100 => 2
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 0101001 => 1001001 => 3
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 0101010 => 1001100 => 3
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 0100100 => 1000100 => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 0010001 => 0010001 => 2
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 0010010 => 0011000 => 2
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 0010100 => 1001000 => 2
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 0001000 => 0010000 => 1
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => 101010010 => 100100110 => 4
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 100100101 => 010001101 => 4
[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => 100010001 => 010000101 => 3
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => 100010010 => 010000110 => 3
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => 100001000 => 010000010 => 2
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => 010001010 => 100001100 => 3
[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => 010000100 => 100000100 => 2
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => 010101001 => 011001001 => 4
[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => 010010001 => 010001001 => 3
[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => 010001000 => 010000100 => 2
[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => 001001001 => 100010001 => 3
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => 001000100 => 100001000 => 2
[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => 001010001 => 010010001 => 3
[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => 001001000 => 010001000 => 2
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => 000100001 => 000100001 => 2
[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => 000100100 => 100010000 => 2
[1,1,1,1,0,1,0,0,0,0] => [[1,2,3,4,6],[5,7,8,9,10]] => 000101000 => 010010000 => 2
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => 000100000 => 1
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Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Map
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.