Identifier
Values
[[1]] => [[1]] => [1] => [1] => 0
[[1,2]] => [[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [[1,2]] => [1,2] => [1,2] => 0
[[1,2,3]] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 1
[[1,2],[3]] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]] => [[1,2],[3]] => [3,1,2] => [3,1,2] => 1
[[1,2,3,4]] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => 1
[[1,2,4],[3]] => [[1,2,3],[4]] => [4,1,2,3] => [4,1,2,3] => 1
[[1,2,3],[4]] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]] => [[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => 1
[[1,2],[3,4]] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,4],[2],[3]] => [[1,2],[3],[4]] => [4,3,1,2] => [4,3,1,2] => 2
[[1,3],[2],[4]] => [[1,2,4],[3]] => [3,1,2,4] => [3,1,2,4] => 1
[[1,2],[3],[4]] => [[1,2,3],[4]] => [4,1,2,3] => [4,1,2,3] => 1
[[1],[2],[3],[4]] => [[1,2],[3],[4]] => [4,3,1,2] => [4,3,1,2] => 2
[[1,2,3,4,5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4],[5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3],[4,5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
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
catabolism
Description
Remove the first row of the standard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry $k$ into tableau $T$ is as follows:
If $k$ is larger than all entries in the first column, place $k$ at the bottom of the first column and the procedure is finished. Otherwise, place $k$ in the first column, replacing the smallest entry, $y$, greater than $k$. Now insert $y$ into the second column using the same procedure: if $y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place $y$ in the second column, replacing, or 'bumping', the smallest entry, $z$, larger than $y$. Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).
Map
to signed permutation
Description
The signed permutation with all signs positive.