Identifier
Values
[1] => [[1]] => [1] => [1] => 1
[2] => [[1,2]] => [1,2] => [1,2] => 1
[1,1] => [[1],[2]] => [2,1] => [2,1] => 3
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 1
[2,1] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 3
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [2,3,1] => 4
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 1
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 3
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [4,3,2,1] => 5
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [2,3,1,4] => 4
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [3,4,1,2] => 5
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 3
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [4,3,2,1,5] => 5
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [2,3,1,4,5] => 4
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,5,4,3,1] => 6
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [3,4,1,2,5] => 5
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [3,4,5,1,2] => 6
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 3
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [4,3,2,1,5,6] => 5
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [2,3,1,4,5,6] => 4
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => 7
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,5,4,3,1,6] => 6
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [3,4,1,2,5,6] => 5
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [4,3,5,6,2,1] => 7
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [3,6,1,5,4,2] => 7
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [3,4,5,1,2,6] => 6
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [4,5,6,1,2,3] => 7
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
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Description
The smallest fixpoint of a permutation.
A fixpoint of a permutation of length $n$ if an index $i$ such that $\pi(i) = i$, and we set $\pi(n+1) = n+1$.
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
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.