Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0010;cc-rep-4
[1,0]=>[[1],[2]]=>[2,1]=>[1,2]=>[.,[.,.]] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[3,1,4,2]=>[[.,.],[[.,.],.]] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,4,3]=>[[.,.],[[.,.],.]] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[5,3,1,6,4,2]=>[[[.,.],.],[[[.,.],.],.]] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[5,2,1,6,4,3]=>[[[.,.],.],[[[.,.],.],.]] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[4,3,1,6,5,2]=>[[[.,.],.],[[[.,.],.],.]] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[4,2,1,6,5,3]=>[[[.,.],.],[[[.,.],.],.]] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[3,2,1,6,5,4]=>[[[.,.],.],[[[.,.],.],.]] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[7,5,3,1,8,6,4,2]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[2,4,7,8,1,3,5,6]=>[7,5,2,1,8,6,4,3]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[2,5,6,8,1,3,4,7]=>[7,4,3,1,8,6,5,2]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[2,5,7,8,1,3,4,6]=>[7,4,2,1,8,6,5,3]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[2,6,7,8,1,3,4,5]=>[7,3,2,1,8,6,5,4]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[3,4,6,8,1,2,5,7]=>[6,5,3,1,8,7,4,2]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[3,4,7,8,1,2,5,6]=>[6,5,2,1,8,7,4,3]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[3,5,7,8,1,2,4,6]=>[6,4,2,1,8,7,5,3]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[4,5,7,8,1,2,3,6]=>[5,4,2,1,8,7,6,3]=>[[[[.,.],.],.],[[[[.,.],.],.],.]] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[4,3,2,1,8,7,6,5]=>[[[[.,.],.],.],[[[[.,.],.],.],.]]
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
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
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.