Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [2,1] => 1
[1,1] => [[1],[2]] => [2,1] => [1,2] => 0
[3] => [[1,2,3]] => [1,2,3] => [3,2,1] => 2
[2,1] => [[1,3],[2]] => [2,1,3] => [2,3,1] => 1
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [4,3,2,1] => 3
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [3,4,2,1] => 2
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [2,1,4,3] => 1
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [2,3,4,1] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [4,5,3,2,1] => 3
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,2,5,4,1] => 2
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,4,5,2,1] => 2
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => 1
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [2,3,4,5,1] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 5
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [5,6,4,3,2,1] => 4
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [4,3,6,5,2,1] => 3
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [4,5,6,3,2,1] => 3
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 3
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [3,5,2,6,4,1] => 2
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [3,4,5,6,2,1] => 2
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [2,1,4,3,6,5] => 1
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [2,4,5,1,6,3] => 1
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [2,3,4,5,6,1] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
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Description
The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.
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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$