Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ 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-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]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[1,2],[3,4],[5,7],[6,8]]=>[6,8,5,7,3,4,1,2]=>[3,1,4,2,6,5,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[1,2],[3,5],[4,6],[7,8]]=>[7,8,4,6,3,5,1,2]=>[2,1,5,3,6,4,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,0,1,1,0,1,0,0]=>[[1,3,4,6],[2,5,7,8]]=>[[1,2],[3,5],[4,7],[6,8]]=>[6,8,4,7,3,5,1,2]=>[3,1,5,2,6,4,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[1,2],[3,6],[4,7],[5,8]]=>[5,8,4,7,3,6,1,2]=>[4,1,5,2,6,3,8,7]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[1,3],[2,4],[5,6],[7,8]]=>[7,8,5,6,2,4,1,3]=>[2,1,4,3,7,5,8,6]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[1,3],[2,4],[5,7],[6,8]]=>[6,8,5,7,2,4,1,3]=>[3,1,4,2,7,5,8,6]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[[1,3],[2,5],[4,6],[7,8]]=>[7,8,4,6,2,5,1,3]=>[2,1,5,3,7,4,8,6]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[[1,3],[2,5],[4,7],[6,8]]=>[6,8,4,7,2,5,1,3]=>[3,1,5,2,7,4,8,6]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[[1,3],[2,6],[4,7],[5,8]]=>[5,8,4,7,2,6,1,3]=>[4,1,5,2,7,3,8,6]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[1,4],[2,5],[3,6],[7,8]]=>[7,8,3,6,2,5,1,4]=>[2,1,6,3,7,4,8,5]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,1,0,0,1,0,0]=>[[1,2,3,6],[4,5,7,8]]=>[[1,4],[2,5],[3,7],[6,8]]=>[6,8,3,7,2,5,1,4]=>[3,1,6,2,7,4,8,5]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[[1,4],[2,6],[3,7],[5,8]]=>[5,8,3,7,2,6,1,4]=>[4,1,6,2,7,3,8,5]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[1,5],[2,6],[3,7],[4,8]]=>[4,8,3,7,2,6,1,5]=>[5,1,6,2,7,3,8,4]=>[[.,.],[[.,.],[[.,.],[[.,.],.]]]] [1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7,10,9]=>[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]] [1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
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
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.