Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0002;cc-rep-4
[1,0]=>[[1],[2]]=>[2,1]=>[2,1]=>[2] [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>[3,2,4,1]=>[3,1] [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>[2,4,3,1]=>[3,1] [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>[3,2,5,4,6,1]=>[4,1,1] [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>[3,2,4,6,5,1]=>[4,1,1] [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>[2,5,3,4,6,1]=>[4,1,1] [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>[2,4,3,6,5,1]=>[4,1,1] [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[2,3,6,4,5,1]=>[4,1,1] [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>[3,2,5,4,7,6,8,1]=>[5,1,1,1] [1,1,0,1,0,1,0,0]=>[[1,2,4,6],[3,5,7,8]]=>[3,5,7,8,1,2,4,6]=>[2,4,3,6,5,8,7,1]=>[5,1,1,1] [1,1,0,1,1,0,0,0]=>[[1,2,4,5],[3,6,7,8]]=>[3,6,7,8,1,2,4,5]=>[2,4,3,5,8,6,7,1]=>[5,1,1,1] [1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[4,5,6,8,1,2,3,7]=>[2,3,7,4,5,6,8,1]=>[5,1,1,1] [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[2,3,4,8,5,6,7,1]=>[5,1,1,1]
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
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
Map
cycle type
Description
The cycle type of a permutation as a partition.