Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1
[1,0]=>[[1],[2]]=>[2,1]=>[1,2]=>[2] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[3,1,4,2]=>[1,2,1] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,1,4,3]=>[1,2,1] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[5,3,1,6,4,2]=>[1,1,2,1,1] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[5,2,1,6,4,3]=>[1,1,2,1,1] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[4,3,1,6,5,2]=>[1,1,2,1,1] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[4,2,1,6,5,3]=>[1,1,2,1,1] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[3,2,1,6,5,4]=>[1,1,2,1,1] [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]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [1,1,0,1,0,0,1,0]=>[[1,2,4,7],[3,5,6,8]]=>[3,5,6,8,1,2,4,7]=>[6,4,3,1,8,7,5,2]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[6,3,2,1,8,7,5,4]=>[1,1,1,2,1,1,1] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[5,4,3,1,8,7,6,2]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [1,1,1,0,1,0,0,0]=>[[1,2,3,5],[4,6,7,8]]=>[4,6,7,8,1,2,3,5]=>[5,3,2,1,8,7,6,4]=>[1,1,1,2,1,1,1] [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]=>[1,1,1,2,1,1,1] [1,1,1,1,1,1,0,0,0,0,0,0]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[7,8,9,10,11,12,1,2,3,4,5,6]=>[6,5,4,3,2,1,12,11,10,9,8,7]=>[1,1,1,1,1,2,1,1,1,1,1]
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
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
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.