Identifier
Values
[1] => [[1]] => {{1}} => [1] => 1
[2] => [[1,2]] => {{1,2}} => [2,1] => 1
[1,1] => [[1],[2]] => {{1},{2}} => [1,2] => 2
[3] => [[1,2,3]] => {{1,2,3}} => [2,3,1] => 1
[2,1] => [[1,3],[2]] => {{1,3},{2}} => [3,2,1] => 1
[1,1,1] => [[1],[2],[3]] => {{1},{2},{3}} => [1,2,3] => 3
[4] => [[1,2,3,4]] => {{1,2,3,4}} => [2,3,4,1] => 1
[3,1] => [[1,3,4],[2]] => {{1,3,4},{2}} => [3,2,4,1] => 1
[2,2] => [[1,2],[3,4]] => {{1,2},{3,4}} => [2,1,4,3] => 2
[2,1,1] => [[1,4],[2],[3]] => {{1,4},{2},{3}} => [4,2,3,1] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => {{1},{2},{3},{4}} => [1,2,3,4] => 4
[5] => [[1,2,3,4,5]] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[4,1] => [[1,3,4,5],[2]] => {{1,3,4,5},{2}} => [3,2,4,5,1] => 1
[3,2] => [[1,2,5],[3,4]] => {{1,2,5},{3,4}} => [2,5,4,3,1] => 1
[3,1,1] => [[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 2
[2,2,1] => [[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 1
[2,1,1,1] => [[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 3
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 5
[6] => [[1,2,3,4,5,6]] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[5,1] => [[1,3,4,5,6],[2]] => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 1
[4,2] => [[1,2,5,6],[3,4]] => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => 1
[4,1,1] => [[1,4,5,6],[2],[3]] => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 2
[3,3] => [[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 2
[3,2,1] => [[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => 1
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => 3
[2,2,2] => [[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => 3
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => {{1,4},{2,6},{3},{5}} => [4,6,3,1,5,2] => 2
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => {{1,6},{2},{3},{4},{5}} => [6,2,3,4,5,1] => 4
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => {{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => 6
[7] => [[1,2,3,4,5,6,7]] => {{1,2,3,4,5,6,7}} => [2,3,4,5,6,7,1] => 1
[6,1] => [[1,3,4,5,6,7],[2]] => {{1,3,4,5,6,7},{2}} => [3,2,4,5,6,7,1] => 1
[5,1,1] => [[1,4,5,6,7],[2],[3]] => {{1,4,5,6,7},{2},{3}} => [4,2,3,5,6,7,1] => 2
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => {{1,5,6,7},{2},{3},{4}} => [5,2,3,4,6,7,1] => 3
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => {{1,6,7},{2},{3},{4},{5}} => [6,2,3,4,5,7,1] => 4
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => {{1},{2},{3},{4},{5},{6},{7}} => [1,2,3,4,5,6,7] => 7
[8] => [[1,2,3,4,5,6,7,8]] => {{1,2,3,4,5,6,7,8}} => [2,3,4,5,6,7,8,1] => 1
[7,1] => [[1,3,4,5,6,7,8],[2]] => {{1,3,4,5,6,7,8},{2}} => [3,2,4,5,6,7,8,1] => 1
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => {{1,4,5,6,7,8},{2},{3}} => [4,2,3,5,6,7,8,1] => 2
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => {{1,5,6,7,8},{2},{3},{4}} => [5,2,3,4,6,7,8,1] => 3
[4,4] => [[1,2,3,4],[5,6,7,8]] => {{1,2,3,4},{5,6,7,8}} => [2,3,4,1,6,7,8,5] => 2
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => {{1,2},{3,4},{5,6},{7,8}} => [2,1,4,3,6,5,8,7] => 4
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => {{1},{2},{3},{4},{5},{6},{7},{8}} => [1,2,3,4,5,6,7,8] => 8
[9] => [[1,2,3,4,5,6,7,8,9]] => {{1,2,3,4,5,6,7,8,9}} => [2,3,4,5,6,7,8,9,1] => 1
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => {{1,3,4,5,6,7,8,9},{2}} => [3,2,4,5,6,7,8,9,1] => 1
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => {{1,4,5,6,7,8,9},{2},{3}} => [4,2,3,5,6,7,8,9,1] => 2
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}} => [1,2,3,4,5,6,7,8,9] => 9
[10] => [[1,2,3,4,5,6,7,8,9,10]] => {{1,2,3,4,5,6,7,8,9,10}} => [2,3,4,5,6,7,8,9,10,1] => 1
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => {{1,3,4,5,6,7,8,9,10},{2}} => [3,2,4,5,6,7,8,9,10,1] => 1
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => {{1,2},{3,4},{5,6},{7,8},{9,10}} => [2,1,4,3,6,5,8,7,10,9] => 5
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}} => [1,2,3,4,5,6,7,8,9,10] => 10
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Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.