Identifier
Values
[1] => [1,0] => [1,0] => [1] => 0
[2] => [1,0,1,0] => [1,1,0,0] => [2,1] => 0
[1,1] => [1,1,0,0] => [1,0,1,0] => [1,2] => 0
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [3,1,2] => 0
[2,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,3,1] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => 2
[2,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,2,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 3
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 4
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => 3
[3,3] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 3
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [5,1,2,6,3,4] => 2
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => 6
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,1,6,2,3] => 5
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => 3
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 2
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,1,3,4,6] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,1,6,2,4] => 4
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,5,2,6,3] => 3
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => 6
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [3,1,5,2,4,6] => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => 5
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,1,4,6] => 2
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,1,3,6,4] => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [4,1,2,5,3,6] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,1,5,6,3] => 3
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,1,6,4] => 3
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,1,5,3,6] => 2
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => 0
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => 3
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => 0
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 1
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 0
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
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
click to show known generating functions       
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
inverse Kreweras complement
Description
Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.