Identifier
Mp00033: to two-row standard tableauStandard tableaux
Mp00084: conjugateStandard tableaux
Mp00069: Permutations complementPermutations
Mp00061: Permutations to increasing tree
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0007;cc-rep-2Cc0010;cc-rep-5
[1,0]=>[[1],[2]]=>[[1,2]]=>[1,2]=>[2,1]=>[[.,.],.] [1,0,1,0]=>[[1,3],[2,4]]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,4,3]=>[[.,.],[[.,.],.]] [1,1,0,0]=>[[1,2],[3,4]]=>[[1,3],[2,4]]=>[2,4,1,3]=>[3,1,4,2]=>[[.,.],[[.,.],.]] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[2,1,4,3,6,5]=>[[.,.],[[.,.],[[.,.],.]]] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2],[3,5],[4,6]]=>[4,6,3,5,1,2]=>[3,1,4,2,6,5]=>[[.,.],[[.,.],[[.,.],.]]] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,3],[2,4],[5,6]]=>[5,6,2,4,1,3]=>[2,1,5,3,6,4]=>[[.,.],[[.,.],[[.,.],.]]] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[[1,3],[2,5],[4,6]]=>[4,6,2,5,1,3]=>[3,1,5,2,6,4]=>[[.,.],[[.,.],[[.,.],.]]] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,4],[2,5],[3,6]]=>[3,6,2,5,1,4]=>[4,1,5,2,6,3]=>[[.,.],[[.,.],[[.,.],.]]] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
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
conjugate
Description
Sends a standard tableau to its conjugate tableau.
Map
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.