Identifier
Values
[1,0] => [[1],[2]] => [[1,2]] => 0 => 0
[1,0,1,0] => [[1,3],[2,4]] => [[1,2,4],[3]] => 010 => 2
[1,1,0,0] => [[1,2],[3,4]] => [[1,2,3,4]] => 000 => 0
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [[1,2,4,6],[3,5]] => 01010 => 6
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [[1,2,4,5,6],[3]] => 01000 => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [[1,2,3,4,6],[5]] => 00010 => 4
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [[1,2,3,5,6],[4]] => 00100 => 3
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [[1,2,3,4,5,6]] => 00000 => 0
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [[1,2,4,6,8],[3,5,7]] => 0101010 => 12
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [[1,2,4,6,7,8],[3,5]] => 0101000 => 6
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [[1,2,4,5,6,8],[3,7]] => 0100010 => 8
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [[1,2,4,5,7,8],[3,6]] => 0100100 => 7
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [[1,2,4,5,6,7,8],[3]] => 0100000 => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [[1,2,3,4,6,8],[5,7]] => 0001010 => 10
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [[1,2,3,4,7,8],[5,6]] => 0001000 => 4
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [[1,2,3,5,6,8],[4,7]] => 0010010 => 9
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [[1,2,3,5,7,8],[4,6]] => 0010100 => 8
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [[1,2,3,5,6,7,8],[4]] => 0010000 => 3
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [[1,2,3,4,5,6,8],[7]] => 0000010 => 6
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [[1,2,3,4,5,7,8],[6]] => 0000100 => 5
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [[1,2,3,4,6,7,8],[5]] => 0001000 => 4
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [[1,2,3,4,5,6,7,8]] => 0000000 => 0
[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => [[1,2,4,6,8,10],[3,5,7,9]] => 010101010 => 20
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [[1,2,4,6,8,9,10],[3,5,7]] => 010101000 => 12
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [[1,2,4,6,7,8,10],[3,5,9]] => 010100010 => 14
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [[1,2,4,6,7,9,10],[3,5,8]] => 010100100 => 13
[1,0,1,0,1,1,1,0,0,0] => [[1,3,5,6,7],[2,4,8,9,10]] => [[1,2,4,6,7,8,9,10],[3,5]] => 010100000 => 6
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [[1,2,4,5,6,8,10],[3,7,9]] => 010001010 => 16
[1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => [[1,2,4,5,6,9,10],[3,7,8]] => 010001000 => 8
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [[1,2,4,5,7,8,10],[3,6,9]] => 010010010 => 15
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => [[1,2,4,5,7,9,10],[3,6,8]] => 010010100 => 14
[1,0,1,1,0,1,1,0,0,0] => [[1,3,4,6,7],[2,5,8,9,10]] => [[1,2,4,5,7,8,9,10],[3,6]] => 010010000 => 7
[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => [[1,2,4,5,6,7,8,10],[3,9]] => 010000010 => 10
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => [[1,2,4,5,6,7,9,10],[3,8]] => 010000100 => 9
[1,0,1,1,1,0,1,0,0,0] => [[1,3,4,5,7],[2,6,8,9,10]] => [[1,2,4,5,6,8,9,10],[3,7]] => 010001000 => 8
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => [[1,2,4,5,6,7,8,9,10],[3]] => 010000000 => 2
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [[1,2,3,4,6,8,10],[5,7,9]] => 000101010 => 18
[1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => [[1,2,3,4,6,9,10],[5,7,8]] => 000101000 => 10
[1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => [[1,2,3,4,7,8,10],[5,6,9]] => 000100010 => 12
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => [[1,2,3,4,7,9,10],[5,6,8]] => 000100100 => 11
[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => [[1,2,3,4,7,8,9,10],[5,6]] => 000100000 => 4
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => [[1,2,3,5,6,8,10],[4,7,9]] => 001001010 => 17
[1,1,0,1,0,0,1,1,0,0] => [[1,2,4,7,8],[3,5,6,9,10]] => [[1,2,3,5,6,9,10],[4,7,8]] => 001001000 => 9
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => [[1,2,3,5,7,8,10],[4,6,9]] => 001010010 => 16
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => [[1,2,3,5,7,9,10],[4,6,8]] => 001010100 => 15
[1,1,0,1,0,1,1,0,0,0] => [[1,2,4,6,7],[3,5,8,9,10]] => [[1,2,3,5,7,8,9,10],[4,6]] => 001010000 => 8
[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => [[1,2,3,5,6,7,8,10],[4,9]] => 001000010 => 11
[1,1,0,1,1,0,0,1,0,0] => [[1,2,4,5,8],[3,6,7,9,10]] => [[1,2,3,5,6,7,9,10],[4,8]] => 001000100 => 10
[1,1,0,1,1,0,1,0,0,0] => [[1,2,4,5,7],[3,6,8,9,10]] => [[1,2,3,5,6,8,9,10],[4,7]] => 001001000 => 9
[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => [[1,2,3,5,6,7,8,9,10],[4]] => 001000000 => 3
[1,1,1,0,0,0,1,0,1,0] => [[1,2,3,7,9],[4,5,6,8,10]] => [[1,2,3,4,5,6,8,10],[7,9]] => 000001010 => 14
[1,1,1,0,0,0,1,1,0,0] => [[1,2,3,7,8],[4,5,6,9,10]] => [[1,2,3,4,5,6,9,10],[7,8]] => 000001000 => 6
[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => [[1,2,3,4,5,7,8,10],[6,9]] => 000010010 => 13
[1,1,1,0,0,1,0,1,0,0] => [[1,2,3,6,8],[4,5,7,9,10]] => [[1,2,3,4,5,7,9,10],[6,8]] => 000010100 => 12
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => [[1,2,3,4,5,8,9,10],[6,7]] => 000010000 => 5
[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => [[1,2,3,4,6,7,8,10],[5,9]] => 000100010 => 12
[1,1,1,0,1,0,0,1,0,0] => [[1,2,3,5,8],[4,6,7,9,10]] => [[1,2,3,4,6,7,9,10],[5,8]] => 000100100 => 11
[1,1,1,0,1,0,1,0,0,0] => [[1,2,3,5,7],[4,6,8,9,10]] => [[1,2,3,4,6,8,9,10],[5,7]] => 000101000 => 10
[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => [[1,2,3,4,6,7,8,9,10],[5]] => 000100000 => 4
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => [[1,2,3,4,5,6,7,8,10],[9]] => 000000010 => 8
[1,1,1,1,0,0,0,1,0,0] => [[1,2,3,4,8],[5,6,7,9,10]] => [[1,2,3,4,5,6,7,9,10],[8]] => 000000100 => 7
[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => [[1,2,3,4,5,6,8,9,10],[7]] => 000001000 => 6
[1,1,1,1,0,1,0,0,0,0] => [[1,2,3,4,6],[5,7,8,9,10]] => [[1,2,3,4,5,7,8,9,10],[6]] => 000010000 => 5
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,2,3,4,5,6,7,8,9,10]] => 000000000 => 0
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Description
The sum of the positions of the ones in a binary word.
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 $1\leq i < n$.
Map
catabolism
Description
Remove the first row of the standard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).