Identifier
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Images
=>
Cc0005;cc-rep-0Cc0007;cc-rep-1Cc0009;cc-rep-3
[1,0]=>[[1],[2]]=>[2,1]=>{{1,2}} [1,0,1,0]=>[[1,3],[2,4]]=>[2,4,1,3]=>{{1,2,4},{3}} [1,1,0,0]=>[[1,2],[3,4]]=>[3,4,1,2]=>{{1,3},{2,4}} [1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[2,4,6,1,3,5]=>{{1,2,4},{3,6},{5}} [1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[2,5,6,1,3,4]=>{{1,2,5},{3,6},{4}} [1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[3,4,6,1,2,5]=>{{1,3,6},{2,4},{5}} [1,1,0,1,0,0]=>[[1,2,4],[3,5,6]]=>[3,5,6,1,2,4]=>{{1,3,6},{2,5},{4}} [1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>{{1,4},{2,5},{3,6}} [1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[2,4,6,8,1,3,5,7]=>{{1,2,4,8},{3,6},{5},{7}} [1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>{{1,5},{2,6},{3,7},{4,8}}
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
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.