Identifier
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Images
=>
Cc0005;cc-rep-1Cc0007;cc-rep-2Cc0007;cc-rep-3
[1]=>[1,0]=>[[1],[2]]=>[[1],[2]] [1,1]=>[1,0,1,0]=>[[1,3],[2,4]]=>[[1,2],[3,4]] [2]=>[1,1,0,0]=>[[1,2],[3,4]]=>[[1,3],[2,4]] [1,1,1]=>[1,0,1,0,1,0]=>[[1,3,5],[2,4,6]]=>[[1,2,4],[3,5,6]] [1,2]=>[1,0,1,1,0,0]=>[[1,3,4],[2,5,6]]=>[[1,2,5],[3,4,6]] [2,1]=>[1,1,0,0,1,0]=>[[1,2,5],[3,4,6]]=>[[1,3,4],[2,5,6]] [3]=>[1,1,1,0,0,0]=>[[1,2,3],[4,5,6]]=>[[1,3,5],[2,4,6]] [1,1,1,1]=>[1,0,1,0,1,0,1,0]=>[[1,3,5,7],[2,4,6,8]]=>[[1,2,4,6],[3,5,7,8]] [1,1,2]=>[1,0,1,0,1,1,0,0]=>[[1,3,5,6],[2,4,7,8]]=>[[1,2,4,7],[3,5,6,8]] [1,2,1]=>[1,0,1,1,0,0,1,0]=>[[1,3,4,7],[2,5,6,8]]=>[[1,2,5,6],[3,4,7,8]] [1,3]=>[1,0,1,1,1,0,0,0]=>[[1,3,4,5],[2,6,7,8]]=>[[1,2,5,7],[3,4,6,8]] [2,1,1]=>[1,1,0,0,1,0,1,0]=>[[1,2,5,7],[3,4,6,8]]=>[[1,3,4,6],[2,5,7,8]] [2,2]=>[1,1,0,0,1,1,0,0]=>[[1,2,5,6],[3,4,7,8]]=>[[1,3,4,7],[2,5,6,8]] [3,1]=>[1,1,1,0,0,0,1,0]=>[[1,2,3,7],[4,5,6,8]]=>[[1,3,5,6],[2,4,7,8]] [4]=>[1,1,1,1,0,0,0,0]=>[[1,2,3,4],[5,6,7,8]]=>[[1,3,5,7],[2,4,6,8]] [1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9],[2,4,6,8,10]]=>[[1,2,4,6,8],[3,5,7,9,10]] [1,1,1,2]=>[1,0,1,0,1,0,1,1,0,0]=>[[1,3,5,7,8],[2,4,6,9,10]]=>[[1,2,4,6,9],[3,5,7,8,10]] [1,1,2,1]=>[1,0,1,0,1,1,0,0,1,0]=>[[1,3,5,6,9],[2,4,7,8,10]]=>[[1,2,4,7,8],[3,5,6,9,10]] [1,1,3]=>[1,0,1,0,1,1,1,0,0,0]=>[[1,3,5,6,7],[2,4,8,9,10]]=>[[1,2,4,7,9],[3,5,6,8,10]] [1,2,1,1]=>[1,0,1,1,0,0,1,0,1,0]=>[[1,3,4,7,9],[2,5,6,8,10]]=>[[1,2,5,6,8],[3,4,7,9,10]] [1,2,2]=>[1,0,1,1,0,0,1,1,0,0]=>[[1,3,4,7,8],[2,5,6,9,10]]=>[[1,2,5,6,9],[3,4,7,8,10]] [1,3,1]=>[1,0,1,1,1,0,0,0,1,0]=>[[1,3,4,5,9],[2,6,7,8,10]]=>[[1,2,5,7,8],[3,4,6,9,10]] [1,4]=>[1,0,1,1,1,1,0,0,0,0]=>[[1,3,4,5,6],[2,7,8,9,10]]=>[[1,2,5,7,9],[3,4,6,8,10]] [2,1,1,1]=>[1,1,0,0,1,0,1,0,1,0]=>[[1,2,5,7,9],[3,4,6,8,10]]=>[[1,3,4,6,8],[2,5,7,9,10]] [2,1,2]=>[1,1,0,0,1,0,1,1,0,0]=>[[1,2,5,7,8],[3,4,6,9,10]]=>[[1,3,4,6,9],[2,5,7,8,10]] [2,2,1]=>[1,1,0,0,1,1,0,0,1,0]=>[[1,2,5,6,9],[3,4,7,8,10]]=>[[1,3,4,7,8],[2,5,6,9,10]] [2,3]=>[1,1,0,0,1,1,1,0,0,0]=>[[1,2,5,6,7],[3,4,8,9,10]]=>[[1,3,4,7,9],[2,5,6,8,10]] [3,1,1]=>[1,1,1,0,0,0,1,0,1,0]=>[[1,2,3,7,9],[4,5,6,8,10]]=>[[1,3,5,6,8],[2,4,7,9,10]] [3,2]=>[1,1,1,0,0,0,1,1,0,0]=>[[1,2,3,7,8],[4,5,6,9,10]]=>[[1,3,5,6,9],[2,4,7,8,10]] [4,1]=>[1,1,1,1,0,0,0,0,1,0]=>[[1,2,3,4,9],[5,6,7,8,10]]=>[[1,3,5,7,8],[2,4,6,9,10]] [5]=>[1,1,1,1,1,0,0,0,0,0]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[[1,3,5,7,9],[2,4,6,8,10]] [1,1,1,1,1,1]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]=>[[1,2,4,6,8,10],[3,5,7,9,11,12]] [1,1,1,1,2]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[[1,3,5,7,9,10],[2,4,6,8,11,12]]=>[[1,2,4,6,8,11],[3,5,7,9,10,12]] [1,1,1,2,1]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[[1,3,5,7,8,11],[2,4,6,9,10,12]]=>[[1,2,4,6,9,10],[3,5,7,8,11,12]] [1,1,1,3]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[[1,3,5,7,8,9],[2,4,6,10,11,12]]=>[[1,2,4,6,9,11],[3,5,7,8,10,12]] [1,1,2,1,1]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[[1,3,5,6,9,11],[2,4,7,8,10,12]]=>[[1,2,4,7,8,10],[3,5,6,9,11,12]] [1,1,2,2]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[[1,3,5,6,9,10],[2,4,7,8,11,12]]=>[[1,2,4,7,8,11],[3,5,6,9,10,12]] [1,1,3,1]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[[1,3,5,6,7,11],[2,4,8,9,10,12]]=>[[1,2,4,7,9,10],[3,5,6,8,11,12]] [1,1,4]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[[1,3,5,6,7,8],[2,4,9,10,11,12]]=>[[1,2,4,7,9,11],[3,5,6,8,10,12]] [1,2,1,1,1]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[[1,3,4,7,9,11],[2,5,6,8,10,12]]=>[[1,2,5,6,8,10],[3,4,7,9,11,12]] [1,2,1,2]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[[1,3,4,7,9,10],[2,5,6,8,11,12]]=>[[1,2,5,6,8,11],[3,4,7,9,10,12]] [1,2,2,1]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[[1,3,4,7,8,11],[2,5,6,9,10,12]]=>[[1,2,5,6,9,10],[3,4,7,8,11,12]] [1,2,3]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[[1,3,4,7,8,9],[2,5,6,10,11,12]]=>[[1,2,5,6,9,11],[3,4,7,8,10,12]] [1,3,1,1]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[[1,3,4,5,9,11],[2,6,7,8,10,12]]=>[[1,2,5,7,8,10],[3,4,6,9,11,12]] [1,3,2]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[[1,3,4,5,9,10],[2,6,7,8,11,12]]=>[[1,2,5,7,8,11],[3,4,6,9,10,12]] [1,4,1]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[[1,3,4,5,6,11],[2,7,8,9,10,12]]=>[[1,2,5,7,9,10],[3,4,6,8,11,12]] [1,5]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[[1,3,4,5,6,7],[2,8,9,10,11,12]]=>[[1,2,5,7,9,11],[3,4,6,8,10,12]] [2,1,1,1,1]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[1,2,5,7,9,11],[3,4,6,8,10,12]]=>[[1,3,4,6,8,10],[2,5,7,9,11,12]] [2,1,1,2]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[1,2,5,7,9,10],[3,4,6,8,11,12]]=>[[1,3,4,6,8,11],[2,5,7,9,10,12]] [2,1,2,1]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[1,2,5,7,8,11],[3,4,6,9,10,12]]=>[[1,3,4,6,9,10],[2,5,7,8,11,12]] [2,1,3]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[[1,2,5,7,8,9],[3,4,6,10,11,12]]=>[[1,3,4,6,9,11],[2,5,7,8,10,12]] [2,2,1,1]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[1,2,5,6,9,11],[3,4,7,8,10,12]]=>[[1,3,4,7,8,10],[2,5,6,9,11,12]] [2,2,2]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[1,2,5,6,9,10],[3,4,7,8,11,12]]=>[[1,3,4,7,8,11],[2,5,6,9,10,12]] [2,3,1]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[[1,2,5,6,7,11],[3,4,8,9,10,12]]=>[[1,3,4,7,9,10],[2,5,6,8,11,12]] [2,4]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[[1,2,5,6,7,8],[3,4,9,10,11,12]]=>[[1,3,4,7,9,11],[2,5,6,8,10,12]] [3,1,1,1]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[[1,2,3,7,9,11],[4,5,6,8,10,12]]=>[[1,3,5,6,8,10],[2,4,7,9,11,12]] [3,1,2]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[[1,2,3,7,9,10],[4,5,6,8,11,12]]=>[[1,3,5,6,8,11],[2,4,7,9,10,12]] [3,2,1]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[[1,2,3,7,8,11],[4,5,6,9,10,12]]=>[[1,3,5,6,9,10],[2,4,7,8,11,12]] [3,3]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[[1,2,3,7,8,9],[4,5,6,10,11,12]]=>[[1,3,5,6,9,11],[2,4,7,8,10,12]] [4,1,1]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[[1,2,3,4,9,11],[5,6,7,8,10,12]]=>[[1,3,5,7,8,10],[2,4,6,9,11,12]] [4,2]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[[1,2,3,4,9,10],[5,6,7,8,11,12]]=>[[1,3,5,7,8,11],[2,4,6,9,10,12]] [5,1]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[[1,2,3,4,5,11],[6,7,8,9,10,12]]=>[[1,3,5,7,9,10],[2,4,6,8,11,12]] [6]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[[1,3,5,7,9,11],[2,4,6,8,10,12]]
Map
bounce path
Description
The bounce path determined by an integer composition.
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
row-to-column-descents
Description
Return a standard tableau whose column descent set equals the row descent set of the original tableau.
A column descent in a standard tableau is an entry $i$ such that $i+1$ appears in a column to the left of the cell containing $i$, in English notation.
A row descent is an entry $i$ such that $i+1$ appears in a row above of the cell containing $i$.