Identifier
Mp00033: to two-row standard tableauStandard tableaux
Mp00089: Permutations Inverse Kreweras complementPermutations
Mp00059: Permutations Robinson-Schensted insertion tableau
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-4
[1,0]=>[[1],[2]]=>[2,1]=>[1,2]=>[[1,2]] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[1,4,2,3]=>[[1,2,3],[4]] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[4,1,2,3]=>[[1,2,3],[4]] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[1,5,2,6,3,4]=>[[1,2,3,4],[5,6]] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[1,5,6,2,3,4]=>[[1,2,3,4],[5,6]] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[5,1,2,6,3,4]=>[[1,2,3,4],[5,6]] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[5,1,6,2,3,4]=>[[1,2,3,4],[5,6]] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[5,6,1,2,3,4]=>[[1,2,3,4],[5,6]] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[1,6,2,7,3,8,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[2,5,6,8,1,3,4,7]=>[1,6,7,2,3,8,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[2,5,7,8,1,3,4,6]=>[1,6,7,2,8,3,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>[1,6,7,8,2,3,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[3,4,6,8,1,2,5,7]=>[6,1,2,7,3,8,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[3,5,6,8,1,2,4,7]=>[6,1,7,2,3,8,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[3,5,7,8,1,2,4,6]=>[6,1,7,2,8,3,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[6,1,7,8,2,3,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[6,7,1,2,3,8,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[6,7,1,8,2,3,4,5]=>[[1,2,3,4,5],[6,7,8]] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[6,7,8,1,2,3,4,5]=>[[1,2,3,4,5],[6,7,8]]
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
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.