Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]
[2]=>[[1,2]]=>[2]
[1,1]=>[[1],[2]]=>[1,1]
[3]=>[[1,2,3]]=>[3]
[2,1]=>[[1,2],[3]]=>[2,1]
[1,1,1]=>[[1],[2],[3]]=>[1,1,1]
[4]=>[[1,2,3,4]]=>[4]
[3,1]=>[[1,2,3],[4]]=>[3,1]
[2,2]=>[[1,2],[3,4]]=>[2,2]
[2,1,1]=>[[1,2],[3],[4]]=>[2,1,1]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[1,1,1,1]
[5]=>[[1,2,3,4,5]]=>[5]
[4,1]=>[[1,2,3,4],[5]]=>[4,1]
[3,2]=>[[1,2,3],[4,5]]=>[3,2]
[3,1,1]=>[[1,2,3],[4],[5]]=>[3,1,1]
[2,2,1]=>[[1,2],[3,4],[5]]=>[2,2,1]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[2,1,1,1]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[1,1,1,1,1]
[6]=>[[1,2,3,4,5,6]]=>[6]
[5,1]=>[[1,2,3,4,5],[6]]=>[5,1]
[4,2]=>[[1,2,3,4],[5,6]]=>[4,2]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[4,1,1]
[3,3]=>[[1,2,3],[4,5,6]]=>[3,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[3,2,1]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[3,1,1,1]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[2,2,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[2,2,1,1]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[2,1,1,1,1]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[1,1,1,1,1,1]
[7]=>[[1,2,3,4,5,6,7]]=>[7]
[6,1]=>[[1,2,3,4,5,6],[7]]=>[6,1]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[5,2]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[5,1,1]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[4,3]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[4,2,1]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[4,1,1,1]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[3,3,1]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[3,2,2]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[3,2,1,1]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[3,1,1,1,1]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[2,2,2,1]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[2,2,1,1,1]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[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]
[8]=>[[1,2,3,4,5,6,7,8]]=>[8]
[7,1]=>[[1,2,3,4,5,6,7],[8]]=>[7,1]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>[6,2]
[6,1,1]=>[[1,2,3,4,5,6],[7],[8]]=>[6,1,1]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[5,3]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[5,2,1]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[5,1,1,1]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[4,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[4,3,1]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[4,2,2]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[4,2,1,1]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[4,1,1,1,1]
[3,3,2]=>[[1,2,3],[4,5,6],[7,8]]=>[3,3,2]
[3,3,1,1]=>[[1,2,3],[4,5,6],[7],[8]]=>[3,3,1,1]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[3,2,2,1]
[3,2,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8]]=>[3,2,1,1,1]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[3,1,1,1,1,1]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[2,2,2,2]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[2,2,2,1,1]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[2,2,1,1,1,1]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[2,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[1,1,1,1,1,1,1,1]
[9]=>[[1,2,3,4,5,6,7,8,9]]=>[9]
[8,1]=>[[1,2,3,4,5,6,7,8],[9]]=>[8,1]
[7,2]=>[[1,2,3,4,5,6,7],[8,9]]=>[7,2]
[7,1,1]=>[[1,2,3,4,5,6,7],[8],[9]]=>[7,1,1]
[6,3]=>[[1,2,3,4,5,6],[7,8,9]]=>[6,3]
[6,2,1]=>[[1,2,3,4,5,6],[7,8],[9]]=>[6,2,1]
[6,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9]]=>[6,1,1,1]
[5,4]=>[[1,2,3,4,5],[6,7,8,9]]=>[5,4]
[5,3,1]=>[[1,2,3,4,5],[6,7,8],[9]]=>[5,3,1]
[5,2,2]=>[[1,2,3,4,5],[6,7],[8,9]]=>[5,2,2]
[5,2,1,1]=>[[1,2,3,4,5],[6,7],[8],[9]]=>[5,2,1,1]
[5,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9]]=>[5,1,1,1,1]
[4,4,1]=>[[1,2,3,4],[5,6,7,8],[9]]=>[4,4,1]
[4,3,2]=>[[1,2,3,4],[5,6,7],[8,9]]=>[4,3,2]
[4,3,1,1]=>[[1,2,3,4],[5,6,7],[8],[9]]=>[4,3,1,1]
[4,2,2,1]=>[[1,2,3,4],[5,6],[7,8],[9]]=>[4,2,2,1]
[4,2,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9]]=>[4,2,1,1,1]
[4,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9]]=>[4,1,1,1,1,1]
[3,3,3]=>[[1,2,3],[4,5,6],[7,8,9]]=>[3,3,3]
[3,3,2,1]=>[[1,2,3],[4,5,6],[7,8],[9]]=>[3,3,2,1]
[3,3,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9]]=>[3,3,1,1,1]
[3,2,2,2]=>[[1,2,3],[4,5],[6,7],[8,9]]=>[3,2,2,2]
[3,2,2,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9]]=>[3,2,2,1,1]
[3,2,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9]]=>[3,2,1,1,1,1]
[3,1,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8],[9]]=>[3,1,1,1,1,1,1]
[2,2,2,2,1]=>[[1,2],[3,4],[5,6],[7,8],[9]]=>[2,2,2,2,1]
[2,2,2,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9]]=>[2,2,2,1,1,1]
[2,2,1,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8],[9]]=>[2,2,1,1,1,1,1]
[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]
[1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9]]=>[1,1,1,1,1,1,1,1,1]
[10]=>[[1,2,3,4,5,6,7,8,9,10]]=>[10]
[9,1]=>[[1,2,3,4,5,6,7,8,9],[10]]=>[9,1]
[8,2]=>[[1,2,3,4,5,6,7,8],[9,10]]=>[8,2]
[8,1,1]=>[[1,2,3,4,5,6,7,8],[9],[10]]=>[8,1,1]
[7,3]=>[[1,2,3,4,5,6,7],[8,9,10]]=>[7,3]
[7,2,1]=>[[1,2,3,4,5,6,7],[8,9],[10]]=>[7,2,1]
[7,1,1,1]=>[[1,2,3,4,5,6,7],[8],[9],[10]]=>[7,1,1,1]
[6,4]=>[[1,2,3,4,5,6],[7,8,9,10]]=>[6,4]
[6,3,1]=>[[1,2,3,4,5,6],[7,8,9],[10]]=>[6,3,1]
[6,2,2]=>[[1,2,3,4,5,6],[7,8],[9,10]]=>[6,2,2]
[6,2,1,1]=>[[1,2,3,4,5,6],[7,8],[9],[10]]=>[6,2,1,1]
[6,1,1,1,1]=>[[1,2,3,4,5,6],[7],[8],[9],[10]]=>[6,1,1,1,1]
[5,5]=>[[1,2,3,4,5],[6,7,8,9,10]]=>[5,5]
[5,4,1]=>[[1,2,3,4,5],[6,7,8,9],[10]]=>[5,4,1]
[5,3,2]=>[[1,2,3,4,5],[6,7,8],[9,10]]=>[5,3,2]
[5,3,1,1]=>[[1,2,3,4,5],[6,7,8],[9],[10]]=>[5,3,1,1]
[5,2,2,1]=>[[1,2,3,4,5],[6,7],[8,9],[10]]=>[5,2,2,1]
[5,2,1,1,1]=>[[1,2,3,4,5],[6,7],[8],[9],[10]]=>[5,2,1,1,1]
[5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[5,1,1,1,1,1]
[4,4,2]=>[[1,2,3,4],[5,6,7,8],[9,10]]=>[4,4,2]
[4,4,1,1]=>[[1,2,3,4],[5,6,7,8],[9],[10]]=>[4,4,1,1]
[4,3,3]=>[[1,2,3,4],[5,6,7],[8,9,10]]=>[4,3,3]
[4,3,2,1]=>[[1,2,3,4],[5,6,7],[8,9],[10]]=>[4,3,2,1]
[4,3,1,1,1]=>[[1,2,3,4],[5,6,7],[8],[9],[10]]=>[4,3,1,1,1]
[4,2,2,2]=>[[1,2,3,4],[5,6],[7,8],[9,10]]=>[4,2,2,2]
[4,2,2,1,1]=>[[1,2,3,4],[5,6],[7,8],[9],[10]]=>[4,2,2,1,1]
[4,2,1,1,1,1]=>[[1,2,3,4],[5,6],[7],[8],[9],[10]]=>[4,2,1,1,1,1]
[4,1,1,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8],[9],[10]]=>[4,1,1,1,1,1,1]
[3,3,3,1]=>[[1,2,3],[4,5,6],[7,8,9],[10]]=>[3,3,3,1]
[3,3,2,2]=>[[1,2,3],[4,5,6],[7,8],[9,10]]=>[3,3,2,2]
[3,3,2,1,1]=>[[1,2,3],[4,5,6],[7,8],[9],[10]]=>[3,3,2,1,1]
[3,3,1,1,1,1]=>[[1,2,3],[4,5,6],[7],[8],[9],[10]]=>[3,3,1,1,1,1]
[3,2,2,2,1]=>[[1,2,3],[4,5],[6,7],[8,9],[10]]=>[3,2,2,2,1]
[3,2,2,1,1,1]=>[[1,2,3],[4,5],[6,7],[8],[9],[10]]=>[3,2,2,1,1,1]
[3,2,1,1,1,1,1]=>[[1,2,3],[4,5],[6],[7],[8],[9],[10]]=>[3,2,1,1,1,1,1]
[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]
[2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[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]
[2,2,2,1,1,1,1]=>[[1,2],[3,4],[5,6],[7],[8],[9],[10]]=>[2,2,2,1,1,1,1]
[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]
[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]
[1,1,1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]=>[1,1,1,1,1,1,1,1,1,1]
[6,6]=>[[1,2,3,4,5,6],[7,8,9,10,11,12]]=>[6,6]
[2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[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 $i$.
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