Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0018;cc-rep-2Cc0018;cc-rep-3
[1,0]=>[[1],[2]]=>[[1,1],[1]]=>[[1,1],[1]] [1,0,1,0]=>[[1,3],[2,4]]=>[[2,2,0,0],[2,1,0],[1,1],[1]]=>[[2,2,0,0],[2,1,0],[1,1],[1]] [1,1,0,0]=>[[1,2],[3,4]]=>[[2,2,0,0],[2,1,0],[2,0],[1]]=>[[2,2,0,0],[2,1,0],[2,0],[1]] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] [1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] [1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] [1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] [1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] [1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] [1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] [1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]=>[[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
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
to Gelfand-Tsetlin pattern
Description
Sends a tableau to its corresponding Gelfand-Tsetlin pattern.
To obtain this Gelfand-Tsetlin pattern, fill in the first row of the pattern with the shape of the tableau.
Then remove the maximal entry from the tableau to obtain a smaller tableau, and repeat the process until the tableau is empty.
Map
Schuetzenberger involution
Description
Applies the Schuetzenberger involution to a Gelfand-Tsetlin pattern.
The Schuetzenberger involution is usually regarded as an involution on semistandard Young tableaux with a fixed bound on the size of the entries. It is also known as evacuation, and in the context of crystal graphs of type $A$ it realizes Lusztig's involution.
In the language of tableaux it is defined as follows. Consider a semistandard tableau with no entry larger than $n$. Use Schuetzenberger's jeu de taquin to slide all entries equal to $1$ to the outer border of the tableau. Do the same for all entries equal to $2$, restricting the tableau to the entries larger than $2$, and so on, until the tableau is a reverse semistandard tableau. Finally, replace each entry with its complement with respect to $n$, that is, replace $e$ with $n+1-e$.