Identifier
Values
[1,0] => [[1],[2]] => [2,1] => [2,1] => 2
[1,0,1,0] => [[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,0] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 2
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => 2
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [4,1,5,6,2,3] => 3
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [4,5,1,2,6,3] => 3
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [4,5,1,6,2,3] => 3
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,5,6,1,2,3] => 2
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,4,6,8,1,3,5,7] => [5,1,6,2,7,3,8,4] => 2
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => 2
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,10,1,2,3,4,5] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [7,8,9,10,11,12,1,2,3,4,5,6] => [7,8,9,10,11,12,1,2,3,4,5,6] => 2
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of distinct diagonal sums of a permutation matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)$$
are $(1,0,1,0,2,0)$, so the statistic is $3$.
Map
inverse
Description
Sends a permutation to its inverse.
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
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.