Identifier
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0002;cc-rep-3
[1]=>[[1]]=>[1]=>[1]
[2]=>[[1,2]]=>[2]=>[2]
[1,1]=>[[1],[2]]=>[1,1]=>[1,1]
[3]=>[[1,2,3]]=>[3]=>[3]
[2,1]=>[[1,3],[2]]=>[1,2]=>[2,1]
[1,1,1]=>[[1],[2],[3]]=>[1,1,1]=>[1,1,1]
[4]=>[[1,2,3,4]]=>[4]=>[4]
[3,1]=>[[1,3,4],[2]]=>[1,3]=>[3,1]
[2,2]=>[[1,2],[3,4]]=>[2,2]=>[2,2]
[2,1,1]=>[[1,4],[2],[3]]=>[1,1,2]=>[2,1,1]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[1,1,1,1]=>[1,1,1,1]
[5]=>[[1,2,3,4,5]]=>[5]=>[5]
[4,1]=>[[1,3,4,5],[2]]=>[1,4]=>[4,1]
[3,2]=>[[1,2,5],[3,4]]=>[2,3]=>[3,2]
[3,1,1]=>[[1,4,5],[2],[3]]=>[1,1,3]=>[3,1,1]
[2,2,1]=>[[1,3],[2,5],[4]]=>[1,2,2]=>[2,2,1]
[2,1,1,1]=>[[1,5],[2],[3],[4]]=>[1,1,1,2]=>[2,1,1,1]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[1,1,1,1,1]=>[1,1,1,1,1]
[6]=>[[1,2,3,4,5,6]]=>[6]=>[6]
[5,1]=>[[1,3,4,5,6],[2]]=>[1,5]=>[5,1]
[4,2]=>[[1,2,5,6],[3,4]]=>[2,4]=>[4,2]
[4,1,1]=>[[1,4,5,6],[2],[3]]=>[1,1,4]=>[4,1,1]
[3,3]=>[[1,2,3],[4,5,6]]=>[3,3]=>[3,3]
[3,2,1]=>[[1,3,6],[2,5],[4]]=>[1,2,3]=>[3,2,1]
[3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>[1,1,1,3]=>[3,1,1,1]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[2,2,2]=>[2,2,2]
[2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>[1,1,2,2]=>[2,2,1,1]
[2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>[1,1,1,1,2]=>[2,1,1,1,1]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,1]=>[1,1,1,1,1,1]
[7]=>[[1,2,3,4,5,6,7]]=>[7]=>[7]
[6,1]=>[[1,3,4,5,6,7],[2]]=>[1,6]=>[6,1]
[5,2]=>[[1,2,5,6,7],[3,4]]=>[2,5]=>[5,2]
[5,1,1]=>[[1,4,5,6,7],[2],[3]]=>[1,1,5]=>[5,1,1]
[4,3]=>[[1,2,3,7],[4,5,6]]=>[3,4]=>[4,3]
[4,2,1]=>[[1,3,6,7],[2,5],[4]]=>[1,2,4]=>[4,2,1]
[4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>[1,1,1,4]=>[4,1,1,1]
[3,3,1]=>[[1,3,4],[2,6,7],[5]]=>[1,3,3]=>[3,3,1]
[3,2,2]=>[[1,2,7],[3,4],[5,6]]=>[2,2,3]=>[3,2,2]
[3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>[1,1,2,3]=>[3,2,1,1]
[3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>[1,1,1,1,3]=>[3,1,1,1,1]
[2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>[1,2,2,2]=>[2,2,2,1]
[2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>[1,1,1,2,2]=>[2,2,1,1,1]
[2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,2]=>[2,1,1,1,1,1]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1]
[8]=>[[1,2,3,4,5,6,7,8]]=>[8]=>[8]
[7,1]=>[[1,3,4,5,6,7,8],[2]]=>[1,7]=>[7,1]
[6,2]=>[[1,2,5,6,7,8],[3,4]]=>[2,6]=>[6,2]
[6,1,1]=>[[1,4,5,6,7,8],[2],[3]]=>[1,1,6]=>[6,1,1]
[5,3]=>[[1,2,3,7,8],[4,5,6]]=>[3,5]=>[5,3]
[5,2,1]=>[[1,3,6,7,8],[2,5],[4]]=>[1,2,5]=>[5,2,1]
[5,1,1,1]=>[[1,5,6,7,8],[2],[3],[4]]=>[1,1,1,5]=>[5,1,1,1]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[4,4]=>[4,4]
[4,3,1]=>[[1,3,4,8],[2,6,7],[5]]=>[1,3,4]=>[4,3,1]
[4,2,2]=>[[1,2,7,8],[3,4],[5,6]]=>[2,2,4]=>[4,2,2]
[4,2,1,1]=>[[1,4,7,8],[2,6],[3],[5]]=>[1,1,2,4]=>[4,2,1,1]
[4,1,1,1,1]=>[[1,6,7,8],[2],[3],[4],[5]]=>[1,1,1,1,4]=>[4,1,1,1,1]
[3,3,2]=>[[1,2,5],[3,4,8],[6,7]]=>[2,3,3]=>[3,3,2]
[3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>[1,1,3,3]=>[3,3,1,1]
[3,2,2,1]=>[[1,3,8],[2,5],[4,7],[6]]=>[1,2,2,3]=>[3,2,2,1]
[3,2,1,1,1]=>[[1,5,8],[2,7],[3],[4],[6]]=>[1,1,1,2,3]=>[3,2,1,1,1]
[3,1,1,1,1,1]=>[[1,7,8],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,3]=>[3,1,1,1,1,1]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[2,2,2,2]=>[2,2,2,2]
[2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5],[7]]=>[1,1,2,2,2]=>[2,2,2,1,1]
[2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[1,1,1,1,2,2]=>[2,2,1,1,1,1]
[2,1,1,1,1,1,1]=>[[1,8],[2],[3],[4],[5],[6],[7]]=>[1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1]
[9]=>[[1,2,3,4,5,6,7,8,9]]=>[9]=>[9]
[8,1]=>[[1,3,4,5,6,7,8,9],[2]]=>[1,8]=>[8,1]
[7,2]=>[[1,2,5,6,7,8,9],[3,4]]=>[2,7]=>[7,2]
[7,1,1]=>[[1,4,5,6,7,8,9],[2],[3]]=>[1,1,7]=>[7,1,1]
[6,3]=>[[1,2,3,7,8,9],[4,5,6]]=>[3,6]=>[6,3]
[6,2,1]=>[[1,3,6,7,8,9],[2,5],[4]]=>[1,2,6]=>[6,2,1]
[6,1,1,1]=>[[1,5,6,7,8,9],[2],[3],[4]]=>[1,1,1,6]=>[6,1,1,1]
[5,4]=>[[1,2,3,4,9],[5,6,7,8]]=>[4,5]=>[5,4]
[5,3,1]=>[[1,3,4,8,9],[2,6,7],[5]]=>[1,3,5]=>[5,3,1]
[5,2,2]=>[[1,2,7,8,9],[3,4],[5,6]]=>[2,2,5]=>[5,2,2]
[5,2,1,1]=>[[1,4,7,8,9],[2,6],[3],[5]]=>[1,1,2,5]=>[5,2,1,1]
[5,1,1,1,1]=>[[1,6,7,8,9],[2],[3],[4],[5]]=>[1,1,1,1,5]=>[5,1,1,1,1]
[4,4,1]=>[[1,3,4,5],[2,7,8,9],[6]]=>[1,4,4]=>[4,4,1]
[4,3,2]=>[[1,2,5,9],[3,4,8],[6,7]]=>[2,3,4]=>[4,3,2]
[4,3,1,1]=>[[1,4,5,9],[2,7,8],[3],[6]]=>[1,1,3,4]=>[4,3,1,1]
[4,2,2,1]=>[[1,3,8,9],[2,5],[4,7],[6]]=>[1,2,2,4]=>[4,2,2,1]
[4,2,1,1,1]=>[[1,5,8,9],[2,7],[3],[4],[6]]=>[1,1,1,2,4]=>[4,2,1,1,1]
[4,1,1,1,1,1]=>[[1,7,8,9],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,4]=>[4,1,1,1,1,1]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[3,3,3]=>[3,3,3]
[3,3,2,1]=>[[1,3,6],[2,5,9],[4,8],[7]]=>[1,2,3,3]=>[3,3,2,1]
[3,3,1,1,1]=>[[1,5,6],[2,8,9],[3],[4],[7]]=>[1,1,1,3,3]=>[3,3,1,1,1]
[3,2,2,2]=>[[1,2,9],[3,4],[5,6],[7,8]]=>[2,2,2,3]=>[3,2,2,2]
[3,2,2,1,1]=>[[1,4,9],[2,6],[3,8],[5],[7]]=>[1,1,2,2,3]=>[3,2,2,1,1]
[3,2,1,1,1,1]=>[[1,6,9],[2,8],[3],[4],[5],[7]]=>[1,1,1,1,2,3]=>[3,2,1,1,1,1]
[3,1,1,1,1,1,1]=>[[1,8,9],[2],[3],[4],[5],[6],[7]]=>[1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1]
[2,2,2,2,1]=>[[1,3],[2,5],[4,7],[6,9],[8]]=>[1,2,2,2,2]=>[2,2,2,2,1]
[2,2,2,1,1,1]=>[[1,5],[2,7],[3,9],[4],[6],[8]]=>[1,1,1,2,2,2]=>[2,2,2,1,1,1]
[2,2,1,1,1,1,1]=>[[1,7],[2,9],[3],[4],[5],[6],[8]]=>[1,1,1,1,1,2,2]=>[2,2,1,1,1,1,1]
[2,1,1,1,1,1,1,1]=>[[1,9],[2],[3],[4],[5],[6],[7],[8]]=>[1,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>[1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1]
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[10]=>[10]
[9,1]=>[[1,3,4,5,6,7,8,9,10],[2]]=>[1,9]=>[9,1]
[8,2]=>[[1,2,5,6,7,8,9,10],[3,4]]=>[2,8]=>[8,2]
[8,1,1]=>[[1,4,5,6,7,8,9,10],[2],[3]]=>[1,1,8]=>[8,1,1]
[7,3]=>[[1,2,3,7,8,9,10],[4,5,6]]=>[3,7]=>[7,3]
[7,2,1]=>[[1,3,6,7,8,9,10],[2,5],[4]]=>[1,2,7]=>[7,2,1]
[7,1,1,1]=>[[1,5,6,7,8,9,10],[2],[3],[4]]=>[1,1,1,7]=>[7,1,1,1]
[6,4]=>[[1,2,3,4,9,10],[5,6,7,8]]=>[4,6]=>[6,4]
[6,3,1]=>[[1,3,4,8,9,10],[2,6,7],[5]]=>[1,3,6]=>[6,3,1]
[6,2,2]=>[[1,2,7,8,9,10],[3,4],[5,6]]=>[2,2,6]=>[6,2,2]
[6,2,1,1]=>[[1,4,7,8,9,10],[2,6],[3],[5]]=>[1,1,2,6]=>[6,2,1,1]
[6,1,1,1,1]=>[[1,6,7,8,9,10],[2],[3],[4],[5]]=>[1,1,1,1,6]=>[6,1,1,1,1]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[5,5]=>[5,5]
[5,4,1]=>[[1,3,4,5,10],[2,7,8,9],[6]]=>[1,4,5]=>[5,4,1]
[5,3,2]=>[[1,2,5,9,10],[3,4,8],[6,7]]=>[2,3,5]=>[5,3,2]
[5,3,1,1]=>[[1,4,5,9,10],[2,7,8],[3],[6]]=>[1,1,3,5]=>[5,3,1,1]
[5,2,2,1]=>[[1,3,8,9,10],[2,5],[4,7],[6]]=>[1,2,2,5]=>[5,2,2,1]
[5,2,1,1,1]=>[[1,5,8,9,10],[2,7],[3],[4],[6]]=>[1,1,1,2,5]=>[5,2,1,1,1]
[5,1,1,1,1,1]=>[[1,7,8,9,10],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,5]=>[5,1,1,1,1,1]
[4,4,2]=>[[1,2,5,6],[3,4,9,10],[7,8]]=>[2,4,4]=>[4,4,2]
[4,4,1,1]=>[[1,4,5,6],[2,8,9,10],[3],[7]]=>[1,1,4,4]=>[4,4,1,1]
[4,3,3]=>[[1,2,3,10],[4,5,6],[7,8,9]]=>[3,3,4]=>[4,3,3]
[4,3,2,1]=>[[1,3,6,10],[2,5,9],[4,8],[7]]=>[1,2,3,4]=>[4,3,2,1]
[4,3,1,1,1]=>[[1,5,6,10],[2,8,9],[3],[4],[7]]=>[1,1,1,3,4]=>[4,3,1,1,1]
[4,2,2,2]=>[[1,2,9,10],[3,4],[5,6],[7,8]]=>[2,2,2,4]=>[4,2,2,2]
[4,2,2,1,1]=>[[1,4,9,10],[2,6],[3,8],[5],[7]]=>[1,1,2,2,4]=>[4,2,2,1,1]
[4,2,1,1,1,1]=>[[1,6,9,10],[2,8],[3],[4],[5],[7]]=>[1,1,1,1,2,4]=>[4,2,1,1,1,1]
[4,1,1,1,1,1,1]=>[[1,8,9,10],[2],[3],[4],[5],[6],[7]]=>[1,1,1,1,1,1,4]=>[4,1,1,1,1,1,1]
[3,3,3,1]=>[[1,3,4],[2,6,7],[5,9,10],[8]]=>[1,3,3,3]=>[3,3,3,1]
[3,3,2,2]=>[[1,2,7],[3,4,10],[5,6],[8,9]]=>[2,2,3,3]=>[3,3,2,2]
[3,3,2,1,1]=>[[1,4,7],[2,6,10],[3,9],[5],[8]]=>[1,1,2,3,3]=>[3,3,2,1,1]
[3,3,1,1,1,1]=>[[1,6,7],[2,9,10],[3],[4],[5],[8]]=>[1,1,1,1,3,3]=>[3,3,1,1,1,1]
[3,2,2,2,1]=>[[1,3,10],[2,5],[4,7],[6,9],[8]]=>[1,2,2,2,3]=>[3,2,2,2,1]
[3,2,2,1,1,1]=>[[1,5,10],[2,7],[3,9],[4],[6],[8]]=>[1,1,1,2,2,3]=>[3,2,2,1,1,1]
[3,2,1,1,1,1,1]=>[[1,7,10],[2,9],[3],[4],[5],[6],[8]]=>[1,1,1,1,1,2,3]=>[3,2,1,1,1,1,1]
[3,1,1,1,1,1,1,1]=>[[1,9,10],[2],[3],[4],[5],[6],[7],[8]]=>[1,1,1,1,1,1,1,3]=>[3,1,1,1,1,1,1,1]
[2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[2,2,2,2,2]=>[2,2,2,2,2]
[2,2,2,2,1,1]=>[[1,4],[2,6],[3,8],[5,10],[7],[9]]=>[1,1,2,2,2,2]=>[2,2,2,2,1,1]
[2,2,2,1,1,1,1]=>[[1,6],[2,8],[3,10],[4],[5],[7],[9]]=>[1,1,1,1,2,2,2]=>[2,2,2,1,1,1,1]
[2,2,1,1,1,1,1,1]=>[[1,8],[2,10],[3],[4],[5],[6],[7],[9]]=>[1,1,1,1,1,1,2,2]=>[2,2,1,1,1,1,1,1]
[2,1,1,1,1,1,1,1,1]=>[[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]=>[1,1,1,1,1,1,1,1,2]=>[2,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[1,1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1,1]
[12]=>[[1,2,3,4,5,6,7,8,9,10,11,12]]=>[12]=>[12]
[6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[6,6]=>[6,6]
[5,5,1,1]=>[[1,4,5,6,7],[2,9,10,11,12],[3],[8]]=>[1,1,5,5]=>[5,5,1,1]
[5,4,3]=>[[1,2,3,7,12],[4,5,6,11],[8,9,10]]=>[3,4,5]=>[5,4,3]
[5,3,2,1,1]=>[[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]=>[1,1,2,3,5]=>[5,3,2,1,1]
[5,2,2,2,1]=>[[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]=>[1,2,2,2,5]=>[5,2,2,2,1]
[4,4,3,1]=>[[1,3,4,8],[2,6,7,12],[5,10,11],[9]]=>[1,3,4,4]=>[4,4,3,1]
[4,4,2,2]=>[[1,2,7,8],[3,4,11,12],[5,6],[9,10]]=>[2,2,4,4]=>[4,4,2,2]
[4,3,3,2]=>[[1,2,5,12],[3,4,8],[6,7,11],[9,10]]=>[2,3,3,4]=>[4,3,3,2]
[3,3,3,2,1]=>[[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]=>[1,2,3,3,3]=>[3,3,3,2,1]
[3,3,2,2,1,1]=>[[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]=>[1,1,2,2,3,3]=>[3,3,2,2,1,1]
[2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[2,2,2,2,2,2]=>[2,2,2,2,2,2]
[1,1,1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]=>[1,1,1,1,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,1,1,1,1]
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
Map
to partition
Description
Sends a composition to the partition obtained by sorting the entries.
searching the database
Sorry, this map was not found in the database.