Identifier
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00151: Permutations to cycle type 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}} [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}} [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
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].