Identifier
Values
[2] => [[1,2]] => [1,2] => {{1},{2}} => 2
[1,1] => [[1],[2]] => [2,1] => {{1,2}} => 0
[3] => [[1,2,3]] => [1,2,3] => {{1},{2},{3}} => 3
[2,1] => [[1,3],[2]] => [2,1,3] => {{1,2},{3}} => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => {{1,3},{2}} => 1
[4] => [[1,2,3,4]] => [1,2,3,4] => {{1},{2},{3},{4}} => 4
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => {{1,2},{3},{4}} => 2
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => {{1,3},{2,4}} => 0
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => {{1,3},{2},{4}} => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => {{1,4},{2,3}} => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => 5
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => 3
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => {{1,3},{2,4},{5}} => 1
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => {{1,3},{2},{4},{5}} => 3
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => {{1,4},{2},{3,5}} => 1
[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] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => {{1,5},{2,4},{3}} => 1
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => 6
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => {{1,2},{3},{4},{5},{6}} => 4
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => {{1,3},{2,4},{5},{6}} => 2
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => {{1,3},{2},{4},{5},{6}} => 4
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}} => 0
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => {{1,4},{2},{3,5},{6}} => 2
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => {{1,4},{2,3},{5},{6}} => 2
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => {{1,5},{2,6},{3},{4}} => 2
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => {{1,5},{2,3},{4,6}} => 0
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => {{1,5},{2,4},{3},{6}} => 2
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}} => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}} => 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
[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => {{1,3},{2,4},{5},{6},{7}} => 3
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => {{1,3},{2},{4},{5},{6},{7}} => 5
[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => {{1,4},{2,5},{3,6},{7}} => 1
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => {{1,4},{2},{3,5},{6},{7}} => 3
[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
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => {{1,5},{2},{3,6},{4,7}} => 1
[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
[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}} => 1
[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}} => 3
[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}} => 1
[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}} => 1
[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] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => {{1,7},{2,6},{3,5},{4}} => 1
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Description
The number of singleton blocks of a set 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
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 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].