Identifier
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1
[1]=>[[1]]=>[1]=>[1]
[2]=>[[1,2]]=>[1,2]=>[1,2]
[1,1]=>[[1],[2]]=>[2,1]=>[2,1]
[3]=>[[1,2,3]]=>[1,2,3]=>[1,2,3]
[2,1]=>[[1,2],[3]]=>[3,1,2]=>[3,1,2]
[1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>[2,3,1]
[4]=>[[1,2,3,4]]=>[1,2,3,4]=>[1,2,3,4]
[3,1]=>[[1,2,3],[4]]=>[4,1,2,3]=>[4,1,2,3]
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>[4,1,3,2]
[2,1,1]=>[[1,2],[3],[4]]=>[4,3,1,2]=>[3,1,4,2]
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>[2,3,4,1]
[5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>[1,2,3,4,5]
[4,1]=>[[1,2,3,4],[5]]=>[5,1,2,3,4]=>[5,1,2,3,4]
[3,2]=>[[1,2,3],[4,5]]=>[4,5,1,2,3]=>[5,1,2,4,3]
[3,1,1]=>[[1,2,3],[4],[5]]=>[5,4,1,2,3]=>[4,1,2,5,3]
[2,2,1]=>[[1,2],[3,4],[5]]=>[5,3,4,1,2]=>[4,1,5,3,2]
[2,1,1,1]=>[[1,2],[3],[4],[5]]=>[5,4,3,1,2]=>[3,1,4,5,2]
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>[2,3,4,5,1]
[6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>[1,2,3,4,5,6]
[5,1]=>[[1,2,3,4,5],[6]]=>[6,1,2,3,4,5]=>[6,1,2,3,4,5]
[4,2]=>[[1,2,3,4],[5,6]]=>[5,6,1,2,3,4]=>[6,1,2,3,5,4]
[4,1,1]=>[[1,2,3,4],[5],[6]]=>[6,5,1,2,3,4]=>[5,1,2,3,6,4]
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>[6,1,2,4,5,3]
[3,2,1]=>[[1,2,3],[4,5],[6]]=>[6,4,5,1,2,3]=>[5,1,2,6,4,3]
[3,1,1,1]=>[[1,2,3],[4],[5],[6]]=>[6,5,4,1,2,3]=>[4,1,2,5,6,3]
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>[4,1,6,3,5,2]
[2,2,1,1]=>[[1,2],[3,4],[5],[6]]=>[6,5,3,4,1,2]=>[4,1,5,3,6,2]
[2,1,1,1,1]=>[[1,2],[3],[4],[5],[6]]=>[6,5,4,3,1,2]=>[3,1,4,5,6,2]
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>[2,3,4,5,6,1]
[7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>[1,2,3,4,5,6,7]
[6,1]=>[[1,2,3,4,5,6],[7]]=>[7,1,2,3,4,5,6]=>[7,1,2,3,4,5,6]
[5,2]=>[[1,2,3,4,5],[6,7]]=>[6,7,1,2,3,4,5]=>[7,1,2,3,4,6,5]
[5,1,1]=>[[1,2,3,4,5],[6],[7]]=>[7,6,1,2,3,4,5]=>[6,1,2,3,4,7,5]
[4,3]=>[[1,2,3,4],[5,6,7]]=>[5,6,7,1,2,3,4]=>[7,1,2,3,5,6,4]
[4,2,1]=>[[1,2,3,4],[5,6],[7]]=>[7,5,6,1,2,3,4]=>[6,1,2,3,7,5,4]
[4,1,1,1]=>[[1,2,3,4],[5],[6],[7]]=>[7,6,5,1,2,3,4]=>[5,1,2,3,6,7,4]
[3,3,1]=>[[1,2,3],[4,5,6],[7]]=>[7,4,5,6,1,2,3]=>[6,1,2,7,4,5,3]
[3,2,2]=>[[1,2,3],[4,5],[6,7]]=>[6,7,4,5,1,2,3]=>[5,1,2,7,4,6,3]
[3,2,1,1]=>[[1,2,3],[4,5],[6],[7]]=>[7,6,4,5,1,2,3]=>[5,1,2,6,4,7,3]
[3,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7]]=>[7,6,5,4,1,2,3]=>[4,1,2,5,6,7,3]
[2,2,2,1]=>[[1,2],[3,4],[5,6],[7]]=>[7,5,6,3,4,1,2]=>[4,1,6,3,7,5,2]
[2,2,1,1,1]=>[[1,2],[3,4],[5],[6],[7]]=>[7,6,5,3,4,1,2]=>[4,1,5,3,6,7,2]
[2,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,1,2]=>[3,1,4,5,6,7,2]
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>[2,3,4,5,6,7,1]
[8]=>[[1,2,3,4,5,6,7,8]]=>[1,2,3,4,5,6,7,8]=>[1,2,3,4,5,6,7,8]
[7,1]=>[[1,2,3,4,5,6,7],[8]]=>[8,1,2,3,4,5,6,7]=>[8,1,2,3,4,5,6,7]
[6,2]=>[[1,2,3,4,5,6],[7,8]]=>[7,8,1,2,3,4,5,6]=>[8,1,2,3,4,5,7,6]
[5,3]=>[[1,2,3,4,5],[6,7,8]]=>[6,7,8,1,2,3,4,5]=>[8,1,2,3,4,6,7,5]
[5,2,1]=>[[1,2,3,4,5],[6,7],[8]]=>[8,6,7,1,2,3,4,5]=>[7,1,2,3,4,8,6,5]
[5,1,1,1]=>[[1,2,3,4,5],[6],[7],[8]]=>[8,7,6,1,2,3,4,5]=>[6,1,2,3,4,7,8,5]
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>[8,1,2,3,5,6,7,4]
[4,3,1]=>[[1,2,3,4],[5,6,7],[8]]=>[8,5,6,7,1,2,3,4]=>[7,1,2,3,8,5,6,4]
[4,2,2]=>[[1,2,3,4],[5,6],[7,8]]=>[7,8,5,6,1,2,3,4]=>[6,1,2,3,8,5,7,4]
[4,2,1,1]=>[[1,2,3,4],[5,6],[7],[8]]=>[8,7,5,6,1,2,3,4]=>[6,1,2,3,7,5,8,4]
[4,1,1,1,1]=>[[1,2,3,4],[5],[6],[7],[8]]=>[8,7,6,5,1,2,3,4]=>[5,1,2,3,6,7,8,4]
[3,2,2,1]=>[[1,2,3],[4,5],[6,7],[8]]=>[8,6,7,4,5,1,2,3]=>[5,1,2,7,4,8,6,3]
[3,1,1,1,1,1]=>[[1,2,3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,1,2,3]=>[4,1,2,5,6,7,8,3]
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,7,2]
[2,2,2,1,1]=>[[1,2],[3,4],[5,6],[7],[8]]=>[8,7,5,6,3,4,1,2]=>[4,1,6,3,7,5,8,2]
[2,2,1,1,1,1]=>[[1,2],[3,4],[5],[6],[7],[8]]=>[8,7,6,5,3,4,1,2]=>[4,1,5,3,6,7,8,2]
[2,1,1,1,1,1,1]=>[[1,2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,1,2]=>[3,1,4,5,6,7,8,2]
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>[2,3,4,5,6,7,8,1]
[5,1,1,1,1,1]=>[[1,2,3,4,5],[6],[7],[8],[9],[10]]=>[10,9,8,7,6,1,2,3,4,5]=>[6,1,2,3,4,7,8,9,10,5]
[2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10]]=>[9,10,7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,10,7,9,2]
[2,2,2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]=>[11,12,9,10,7,8,5,6,3,4,1,2]=>[4,1,6,3,8,5,10,7,12,9,11,2]
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
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
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
- $des$ is the number of descents, St000021The number of descents of a permutation.,
- $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
- $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
- $Ebot$ is the sum of the excedance bottoms,
- $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
- $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
- $Res$ is the sum of the (right) embracing numbers,
- $Ine$ is the sum of the side numbers.
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