Identifier
Values
[1,0] => [1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0] => [1,1,0,1,0,0] => [3,1,2] => [3,2,1] => 0
[1,1,0,0] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => [3,1,4,2] => 1
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,2,1,4] => 0
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,3,2] => 0
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,3,2,1] => 0
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,2,4,1,3] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [3,1,4,2,5] => 1
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [5,4,2,1,3] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,4,2,5,3] => 1
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [4,2,1,5,3] => 1
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [5,3,1,4,2] => 1
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,3,2,1,5] => 0
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,4,3] => 0
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,4,3,2] => 0
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,4,3,2,1] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [5,1,3,6,2,4] => 2
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => [5,2,4,1,3,6] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [3,4,1,6,2,5] => [3,1,6,5,2,4] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [3,4,6,1,2,5] => [6,5,2,4,1,3] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => [3,1,4,2,5,6] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [3,1,5,6,2,4] => [5,2,1,3,6,4] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4,6] => [5,4,2,1,3,6] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [3,5,6,1,2,4] => [5,2,6,4,1,3] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4,6] => [3,1,5,4,2,6] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [3,1,2,6,4,5] => [3,2,1,6,5,4] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [3,6,1,2,4,5] => [3,1,6,5,4,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => [1,6,3,5,2,4] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => [1,4,2,5,3,6] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => [1,6,5,3,2,4] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,6,2,3,5] => [1,4,2,6,5,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => [1,4,3,2,5,6] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => [6,3,5,2,1,4] => 2
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [4,1,5,2,3,6] => [4,2,1,5,3,6] => 1
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,5,1,6,2,3] => [5,2,6,3,1,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 3
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => [5,3,1,4,2,6] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => [4,2,1,6,5,3] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => [6,5,3,1,4,2] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => [4,3,2,1,5,6] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => [1,2,5,3,6,4] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => [1,2,5,4,3,6] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => [1,5,3,2,6,4] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,6,2,3,4] => [1,6,4,2,5,3] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => [1,5,4,3,2,6] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => [5,3,2,1,6,4] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [5,1,6,2,3,4] => [6,4,2,1,5,3] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => [5,4,3,2,1,6] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => [1,2,6,5,4,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[] => [1,0] => [1] => [1] => 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
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.