Identifier
Values
[1,0] => [[1],[2]] => [[1,2]] => [1,2] => 0
[1,0,1,0] => [[1,3],[2,4]] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,0,0] => [[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => 1
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 0
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [[1,2],[3,5],[4,6]] => [4,6,3,5,1,2] => 1
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [[1,3],[2,4],[5,6]] => [5,6,2,4,1,3] => 1
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => 3
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => 4
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Description
The number of cyclic alignments of a permutation.
The pair $(i,j)$ is a cyclic alignment of a permutation $\pi$ if $i, j, \pi(j), \pi(i)$ are cyclically ordered and all distinct, see Section 5 of [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
conjugate
Description
Sends a standard tableau to its conjugate tableau.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.