Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => [2,1] => 0
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[2,1] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 0
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 0
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 1
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 0
[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 occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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.