Identifier
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Images
=>
Cc0002;cc-rep-0Cc0007;cc-rep-1Cc0009;cc-rep-3
[1]=>[[1]]=>[1]=>{{1}}
[2]=>[[1,2]]=>[1,2]=>{{1},{2}}
[1,1]=>[[1],[2]]=>[2,1]=>{{1,2}}
[3]=>[[1,2,3]]=>[1,2,3]=>{{1},{2},{3}}
[2,1]=>[[1,3],[2]]=>[2,1,3]=>{{1,2},{3}}
[1,1,1]=>[[1],[2],[3]]=>[3,2,1]=>{{1,3},{2}}
[4]=>[[1,2,3,4]]=>[1,2,3,4]=>{{1},{2},{3},{4}}
[3,1]=>[[1,3,4],[2]]=>[2,1,3,4]=>{{1,2},{3},{4}}
[2,2]=>[[1,2],[3,4]]=>[3,4,1,2]=>{{1,3},{2,4}}
[2,1,1]=>[[1,4],[2],[3]]=>[3,2,1,4]=>{{1,3},{2},{4}}
[1,1,1,1]=>[[1],[2],[3],[4]]=>[4,3,2,1]=>{{1,4},{2,3}}
[5]=>[[1,2,3,4,5]]=>[1,2,3,4,5]=>{{1},{2},{3},{4},{5}}
[4,1]=>[[1,3,4,5],[2]]=>[2,1,3,4,5]=>{{1,2},{3},{4},{5}}
[3,2]=>[[1,2,5],[3,4]]=>[3,4,1,2,5]=>{{1,3},{2,4},{5}}
[3,1,1]=>[[1,4,5],[2],[3]]=>[3,2,1,4,5]=>{{1,3},{2},{4},{5}}
[2,2,1]=>[[1,3],[2,5],[4]]=>[4,2,5,1,3]=>{{1,4},{2},{3,5}}
[2,1,1,1]=>[[1,5],[2],[3],[4]]=>[4,3,2,1,5]=>{{1,4},{2,3},{5}}
[1,1,1,1,1]=>[[1],[2],[3],[4],[5]]=>[5,4,3,2,1]=>{{1,5},{2,4},{3}}
[6]=>[[1,2,3,4,5,6]]=>[1,2,3,4,5,6]=>{{1},{2},{3},{4},{5},{6}}
[5,1]=>[[1,3,4,5,6],[2]]=>[2,1,3,4,5,6]=>{{1,2},{3},{4},{5},{6}}
[4,2]=>[[1,2,5,6],[3,4]]=>[3,4,1,2,5,6]=>{{1,3},{2,4},{5},{6}}
[4,1,1]=>[[1,4,5,6],[2],[3]]=>[3,2,1,4,5,6]=>{{1,3},{2},{4},{5},{6}}
[3,3]=>[[1,2,3],[4,5,6]]=>[4,5,6,1,2,3]=>{{1,4},{2,5},{3,6}}
[3,2,1]=>[[1,3,6],[2,5],[4]]=>[4,2,5,1,3,6]=>{{1,4},{2},{3,5},{6}}
[3,1,1,1]=>[[1,5,6],[2],[3],[4]]=>[4,3,2,1,5,6]=>{{1,4},{2,3},{5},{6}}
[2,2,2]=>[[1,2],[3,4],[5,6]]=>[5,6,3,4,1,2]=>{{1,5},{2,6},{3},{4}}
[2,2,1,1]=>[[1,4],[2,6],[3],[5]]=>[5,3,2,6,1,4]=>{{1,5},{2,3},{4,6}}
[2,1,1,1,1]=>[[1,6],[2],[3],[4],[5]]=>[5,4,3,2,1,6]=>{{1,5},{2,4},{3},{6}}
[1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1]=>{{1,6},{2,5},{3,4}}
[7]=>[[1,2,3,4,5,6,7]]=>[1,2,3,4,5,6,7]=>{{1},{2},{3},{4},{5},{6},{7}}
[6,1]=>[[1,3,4,5,6,7],[2]]=>[2,1,3,4,5,6,7]=>{{1,2},{3},{4},{5},{6},{7}}
[5,2]=>[[1,2,5,6,7],[3,4]]=>[3,4,1,2,5,6,7]=>{{1,3},{2,4},{5},{6},{7}}
[5,1,1]=>[[1,4,5,6,7],[2],[3]]=>[3,2,1,4,5,6,7]=>{{1,3},{2},{4},{5},{6},{7}}
[4,3]=>[[1,2,3,7],[4,5,6]]=>[4,5,6,1,2,3,7]=>{{1,4},{2,5},{3,6},{7}}
[4,2,1]=>[[1,3,6,7],[2,5],[4]]=>[4,2,5,1,3,6,7]=>{{1,4},{2},{3,5},{6},{7}}
[4,1,1,1]=>[[1,5,6,7],[2],[3],[4]]=>[4,3,2,1,5,6,7]=>{{1,4},{2,3},{5},{6},{7}}
[3,3,1]=>[[1,3,4],[2,6,7],[5]]=>[5,2,6,7,1,3,4]=>{{1,5},{2},{3,6},{4,7}}
[3,2,2]=>[[1,2,7],[3,4],[5,6]]=>[5,6,3,4,1,2,7]=>{{1,5},{2,6},{3},{4},{7}}
[3,2,1,1]=>[[1,4,7],[2,6],[3],[5]]=>[5,3,2,6,1,4,7]=>{{1,5},{2,3},{4,6},{7}}
[3,1,1,1,1]=>[[1,6,7],[2],[3],[4],[5]]=>[5,4,3,2,1,6,7]=>{{1,5},{2,4},{3},{6},{7}}
[2,2,2,1]=>[[1,3],[2,5],[4,7],[6]]=>[6,4,7,2,5,1,3]=>{{1,6},{2,4},{3,7},{5}}
[2,2,1,1,1]=>[[1,5],[2,7],[3],[4],[6]]=>[6,4,3,2,7,1,5]=>{{1,6},{2,4},{3},{5,7}}
[2,1,1,1,1,1]=>[[1,7],[2],[3],[4],[5],[6]]=>[6,5,4,3,2,1,7]=>{{1,6},{2,5},{3,4},{7}}
[1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7]]=>[7,6,5,4,3,2,1]=>{{1,7},{2,6},{3,5},{4}}
[4,4]=>[[1,2,3,4],[5,6,7,8]]=>[5,6,7,8,1,2,3,4]=>{{1,5},{2,6},{3,7},{4,8}}
[3,3,1,1]=>[[1,4,5],[2,7,8],[3],[6]]=>[6,3,2,7,8,1,4,5]=>{{1,6},{2,3},{4,7},{5,8}}
[2,2,2,2]=>[[1,2],[3,4],[5,6],[7,8]]=>[7,8,5,6,3,4,1,2]=>{{1,7},{2,8},{3,5},{4,6}}
[2,2,1,1,1,1]=>[[1,6],[2,8],[3],[4],[5],[7]]=>[7,5,4,3,2,8,1,6]=>{{1,7},{2,5},{3,4},{6,8}}
[1,1,1,1,1,1,1,1]=>[[1],[2],[3],[4],[5],[6],[7],[8]]=>[8,7,6,5,4,3,2,1]=>{{1,8},{2,7},{3,6},{4,5}}
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
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
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
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