Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[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]=>[[1,2],[3]]=>[2,1]=>[1,2]
[1,1,1]=>[[1],[2],[3]]=>[1,1,1]=>[1,1,1]
[4]=>[[1,2,3,4]]=>[4]=>[4]
[3,1]=>[[1,2,3],[4]]=>[3,1]=>[1,3]
[2,2]=>[[1,2],[3,4]]=>[2,2]=>[2,2]
[2,1,1]=>[[1,2],[3],[4]]=>[2,1,1]=>[1,1,2]
[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]=>[[1,2,3,4],[5]]=>[4,1]=>[1,4]
[3,2]=>[[1,2,3],[4,5]]=>[3,2]=>[2,3]
[3,1,1]=>[[1,2,3],[4],[5]]=>[3,1,1]=>[1,1,3]
[2,2,1]=>[[1,2],[3,4],[5]]=>[2,2,1]=>[1,2,2]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[2,1,1,1]=>[1,1,1,2]
[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]=>[[1,2,3,4,5],[6]]=>[5,1]=>[1,5]
[4,2]=>[[1,2,3,4],[5,6]]=>[4,2]=>[2,4]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[4,1,1]=>[1,1,4]
[3,3]=>[[1,2,3],[4,5,6]]=>[3,3]=>[3,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[3,2,1]=>[1,2,3]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[3,1,1,1]=>[1,1,1,3]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[2,2,2]=>[2,2,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[2,2,1,1]=>[1,1,2,2]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[2,1,1,1,1]=>[1,1,1,1,2]
[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]=>[[1,2,3,4,5,6],[7]]=>[6,1]=>[1,6]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[5,2]=>[2,5]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[5,1,1]=>[1,1,5]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[4,3]=>[3,4]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[4,2,1]=>[1,2,4]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[4,1,1,1]=>[1,1,1,4]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[3,3,1]=>[1,3,3]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[3,2,2]=>[2,2,3]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[3,2,1,1]=>[1,1,2,3]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[3,1,1,1,1]=>[1,1,1,1,3]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[2,2,2,1]=>[1,2,2,2]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[2,2,1,1,1]=>[1,1,1,2,2]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[2,1,1,1,1,1]=>[1,1,1,1,1,2]
[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]=>[[1,2,3,4,5,6,7],[8]]=>[7,1]=>[1,7]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>[6,2]=>[2,6]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[6,1,1]=>[1,1,6]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[5,3]=>[3,5]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[5,2,1]=>[1,2,5]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[5,1,1,1]=>[1,1,1,5]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[4,4]=>[4,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[4,3,1]=>[1,3,4]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[4,2,2]=>[2,2,4]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[4,2,1,1]=>[1,1,2,4]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[4,1,1,1,1]=>[1,1,1,1,4]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[3,3,2]=>[2,3,3]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[3,3,1,1]=>[1,1,3,3]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[3,2,2,1]=>[1,2,2,3]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[3,2,1,1,1]=>[1,1,1,2,3]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[3,1,1,1,1,1]=>[1,1,1,1,1,3]
[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]=>[[1,2],[3,4],[5,6],[7],[8]]=>[2,2,2,1,1]=>[1,1,2,2,2]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[2,2,1,1,1,1]=>[1,1,1,1,2,2]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2]
[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]=>[[1,2,3,4,5,6,7,8],[9]]=>[8,1]=>[1,8]
[7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[7,2]=>[2,7]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[7,1,1]=>[1,1,7]
[6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[6,3]=>[3,6]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[6,2,1]=>[1,2,6]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[6,1,1,1]=>[1,1,1,6]
[5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[5,4]=>[4,5]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[5,3,1]=>[1,3,5]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[5,2,2]=>[2,2,5]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[5,2,1,1]=>[1,1,2,5]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[5,1,1,1,1]=>[1,1,1,1,5]
[4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[4,4,1]=>[1,4,4]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[4,3,2]=>[2,3,4]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[4,3,1,1]=>[1,1,3,4]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[4,2,2,1]=>[1,2,2,4]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[4,2,1,1,1]=>[1,1,1,2,4]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[4,1,1,1,1,1]=>[1,1,1,1,1,4]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[3,3,3]=>[3,3,3]
[3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[3,3,2,1]=>[1,2,3,3]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[3,3,1,1,1]=>[1,1,1,3,3]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[3,2,2,2]=>[2,2,2,3]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[3,2,2,1,1]=>[1,1,2,2,3]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[3,2,1,1,1,1]=>[1,1,1,1,2,3]
[3,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9]]=>[3,1,1,1,1,1,1]=>[1,1,1,1,1,1,3]
[2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[2,2,2,2,1]=>[1,2,2,2,2]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[2,2,2,1,1,1]=>[1,1,1,2,2,2]
[2,2,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9]]=>[2,2,1,1,1,1,1]=>[1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9]]=>[2,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,2]
[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]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[9,1]=>[1,9]
[8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[8,2]=>[2,8]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[8,1,1]=>[1,1,8]
[7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[7,3]=>[3,7]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[7,2,1]=>[1,2,7]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[7,1,1,1]=>[1,1,1,7]
[6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[6,4]=>[4,6]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[6,3,1]=>[1,3,6]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[6,2,2]=>[2,2,6]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[6,2,1,1]=>[1,1,2,6]
[6,1,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9],[10]]=>[6,1,1,1,1]=>[1,1,1,1,6]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[5,5]=>[5,5]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[5,4,1]=>[1,4,5]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[5,3,2]=>[2,3,5]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[5,3,1,1]=>[1,1,3,5]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[5,2,2,1]=>[1,2,2,5]
[5,2,1,1,1]=>[[1,2,3,4,5],[6,7],[8],[9],[10]]=>[5,2,1,1,1]=>[1,1,1,2,5]
[5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[5,1,1,1,1,1]=>[1,1,1,1,1,5]
[4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[4,4,2]=>[2,4,4]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[4,4,1,1]=>[1,1,4,4]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[4,3,3]=>[3,3,4]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[4,3,2,1]=>[1,2,3,4]
[4,3,1,1,1]=>[[1,2,3,4],[5,6,7],[8],[9],[10]]=>[4,3,1,1,1]=>[1,1,1,3,4]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[4,2,2,2]=>[2,2,2,4]
[4,2,2,1,1]=>[[1,2,3,4],[5,6],[7,8],[9],[10]]=>[4,2,2,1,1]=>[1,1,2,2,4]
[4,2,1,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]]=>[4,2,1,1,1,1]=>[1,1,1,1,2,4]
[4,1,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]]=>[4,1,1,1,1,1,1]=>[1,1,1,1,1,1,4]
[3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[3,3,3,1]=>[1,3,3,3]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[3,3,2,2]=>[2,2,3,3]
[3,3,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9],[10]]=>[3,3,2,1,1]=>[1,1,2,3,3]
[3,3,1,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]]=>[3,3,1,1,1,1]=>[1,1,1,1,3,3]
[3,2,2,2,1]=>[[1,2,3],[4,5],[6,7],[8,9],[10]]=>[3,2,2,2,1]=>[1,2,2,2,3]
[3,2,2,1,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]]=>[3,2,2,1,1,1]=>[1,1,1,2,2,3]
[3,2,1,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]]=>[3,2,1,1,1,1,1]=>[1,1,1,1,1,2,3]
[3,1,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9],[10]]=>[3,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,3]
[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]=>[[1,2],[3,4],[5,6],[7,8],[9],[10]]=>[2,2,2,2,1,1]=>[1,1,2,2,2,2]
[2,2,2,1,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]]=>[2,2,2,1,1,1,1]=>[1,1,1,1,2,2,2]
[2,2,1,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9],[10]]=>[2,2,1,1,1,1,1,1]=>[1,1,1,1,1,1,2,2]
[2,1,1,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[2,1,1,1,1,1,1,1,1]=>[1,1,1,1,1,1,1,1,2]
[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]
[4,4,3]=>[[1,2,3,4],[5,6,7,8],[9,10,11]]=>[4,4,3]=>[3,4,4]
[3,3,3,2]=>[[1,2,3],[4,5,6],[7,8,9],[10,11]]=>[3,3,3,2]=>[2,3,3,3]
[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]
[6,4,2]=>[[1,2,3,4,5,6],[7,8,9,10],[11,12]]=>[6,4,2]=>[2,4,6]
[5,4,3]=>[[1,2,3,4,5],[6,7,8,9],[10,11,12]]=>[5,4,3]=>[3,4,5]
[5,4,2,1]=>[[1,2,3,4,5],[6,7,8,9],[10,11],[12]]=>[5,4,2,1]=>[1,2,4,5]
[5,3,3,1]=>[[1,2,3,4,5],[6,7,8],[9,10,11],[12]]=>[5,3,3,1]=>[1,3,3,5]
[5,3,2,2]=>[[1,2,3,4,5],[6,7,8],[9,10],[11,12]]=>[5,3,2,2]=>[2,2,3,5]
[4,4,3,1]=>[[1,2,3,4],[5,6,7,8],[9,10,11],[12]]=>[4,4,3,1]=>[1,3,4,4]
[4,4,2,2]=>[[1,2,3,4],[5,6,7,8],[9,10],[11,12]]=>[4,4,2,2]=>[2,2,4,4]
[4,3,3,2]=>[[1,2,3,4],[5,6,7],[8,9,10],[11,12]]=>[4,3,3,2]=>[2,3,3,4]
[4,3,2,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]=>[4,3,2,2,1]=>[1,2,2,3,4]
[4,2,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]=>[4,2,2,2,2]=>[2,2,2,2,4]
[3,3,3,2,1]=>[[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]=>[3,3,3,2,1]=>[1,2,3,3,3]
[3,3,2,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9,10],[11],[12]]=>[3,3,2,2,1,1]=>[1,1,2,2,3,3]
[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]
[2,2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]=>[2,2,2,2,2,2,2]=>[2,2,2,2,2,2,2]
[2,2,2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]=>[2,2,2,2,2,2,2,2]=>[2,2,2,2,2,2,2,2]
[]=>[]=>[0]=>[0]
[2,2,2,2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]=>[2,2,2,2,2,2,2,2,2]=>[2,2,2,2,2,2,2,2,2]
[2,2,2,2,2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20]]=>[2,2,2,2,2,2,2,2,2,2]=>[2,2,2,2,2,2,2,2,2,2]
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
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
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
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