Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-2Cc0007;cc-rep-3
[1,0]=>[[1],[2]]=>[[1,2]]=>[[1,2]] [1,0,1,0]=>[[1,3],[2,4]]=>[[1,2,4],[3]]=>[[1,2,4],[3]] [1,1,0,0]=>[[1,2],[3,4]]=>[[1,2,3,4]]=>[[1,2,3,4]] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2,4,6],[3,5]]=>[[1,2,4,6],[3,5]] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2,4,5,6],[3]]=>[[1,2,3,4,6],[5]] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,2,3,4,6],[5]]=>[[1,2,4,5,6],[3]] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[1,2,3,5,6],[4]]=>[[1,2,3,5,6],[4]] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,2,3,4,5,6]]=>[[1,2,3,4,5,6]] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2,4,6,8],[3,5,7]]=>[[1,2,4,6,8],[3,5,7]] [1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[1,2,4,6,7,8],[3,5]]=>[[1,2,3,4,6,8],[5,7]] [1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[1,2,4,5,6,8],[3,7]]=>[[1,2,4,5,6,8],[3,7]] [1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[[1,2,4,5,7,8],[3,6]]=>[[1,2,3,5,6,8],[4,7]] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[1,2,4,5,6,7,8],[3]]=>[[1,2,3,4,5,6,8],[7]] [1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[1,2,3,4,6,8],[5,7]]=>[[1,2,4,6,7,8],[3,5]] [1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[1,2,3,4,7,8],[5,6]]=>[[1,2,3,4,7,8],[5,6]] [1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[[1,2,3,5,6,8],[4,7]]=>[[1,2,4,5,7,8],[3,6]] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[[1,2,3,5,7,8],[4,6]]=>[[1,2,3,5,7,8],[4,6]] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[[1,2,3,5,6,7,8],[4]]=>[[1,2,3,4,5,7,8],[6]] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[1,2,3,4,5,6,8],[7]]=>[[1,2,4,5,6,7,8],[3]] [1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[[1,2,3,4,5,7,8],[6]]=>[[1,2,3,5,6,7,8],[4]] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[[1,2,3,4,6,7,8],[5]]=>[[1,2,3,4,6,7,8],[5]] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[1,2,3,4,5,6,7,8]]=>[[1,2,3,4,5,6,7,8]] [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]]=>[[1,2,3,4,5,6,7,8,9,10]]
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
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).
Map
Schützenberger involution
Description
Sends a standard tableau to the standard tableau obtained via the Schützenberger involution.