- FindStat

Identifier

Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000039: Permutations ⟶ ℤ

Values

[[1]] => [1] => 0

[[1,0],[0,1]] => [1,2] => 0

[[0,1],[1,0]] => [2,1] => 0

[[1,0,0],[0,1,0],[0,0,1]] => [1,2,3] => 0

[[0,1,0],[1,0,0],[0,0,1]] => [2,1,3] => 0

[[1,0,0],[0,0,1],[0,1,0]] => [1,3,2] => 0

[[0,1,0],[1,-1,1],[0,1,0]] => [1,3,2] => 0

[[0,0,1],[1,0,0],[0,1,0]] => [3,1,2] => 0

[[0,1,0],[0,0,1],[1,0,0]] => [2,3,1] => 1

[[0,0,1],[0,1,0],[1,0,0]] => [3,2,1] => 0

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [1,2,3,4] => 0

[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [2,1,3,4] => 0

[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [1,3,2,4] => 0

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [1,3,2,4] => 0

[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [3,1,2,4] => 0

[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [2,3,1,4] => 1

[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [3,2,1,4] => 0

[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [1,2,4,3] => 0

[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [2,1,4,3] => 0

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [1,2,4,3] => 0

[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [1,2,4,3] => 0

[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [2,1,4,3] => 0

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [2,1,4,3] => 0

[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [2,1,4,3] => 0

[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [1,4,2,3] => 0

[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [1,4,2,3] => 0

[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [1,4,2,3] => 0

[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [4,1,2,3] => 0

[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [2,4,1,3] => 1

[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [2,4,1,3] => 1

[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => [4,2,1,3] => 0

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [1,3,4,2] => 1

[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [1,3,4,2] => 1

[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [3,1,4,2] => 1

[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => [1,3,4,2] => 1

[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [3,1,4,2] => 1

[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 0

[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 0

[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 0

[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [4,1,3,2] => 0

[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [1,4,3,2] => 0

[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [1,4,3,2] => 0

[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [4,1,3,2] => 0

[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [3,4,1,2] => 2

[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => [4,3,1,2] => 1

[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [2,3,4,1] => 2

[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [3,2,4,1] => 1

[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => [2,4,3,1] => 1

[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [2,4,3,1] => 1

[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [4,2,3,1] => 0

[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => [3,4,2,1] => 1

[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [4,3,2,1] => 0

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [1,2,3,4,5] => 0

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [2,1,3,4,5] => 0

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [1,3,2,4,5] => 0

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [1,3,2,4,5] => 0

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [3,1,2,4,5] => 0

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [2,3,1,4,5] => 1

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [3,2,1,4,5] => 0

[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,2,4,3,5] => 0

[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 0

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,2,4,3,5] => 0

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,2,4,3,5] => 0

[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 0

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 0

[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 0

[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,4,2,3,5] => 0

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,4,2,3,5] => 0

[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,4,2,3,5] => 0

[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [4,1,2,3,5] => 0

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,4,1,3,5] => 1

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,4,1,3,5] => 1

[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [4,2,1,3,5] => 0

[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,3,4,2,5] => 1

[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,3,4,2,5] => 1

[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [3,1,4,2,5] => 1

[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,3,4,2,5] => 1

[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [3,1,4,2,5] => 1

[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 0

[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 0

[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 0

[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [4,1,3,2,5] => 0

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 0

[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 0

[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [4,1,3,2,5] => 0

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [3,4,1,2,5] => 2

[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [4,3,1,2,5] => 1

[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [2,3,4,1,5] => 2

[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [3,2,4,1,5] => 1

[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [2,4,3,1,5] => 1

[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [2,4,3,1,5] => 1

[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [4,2,3,1,5] => 0

[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [3,4,2,1,5] => 1

[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [4,3,2,1,5] => 0

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [1,2,3,5,4] => 0

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [2,1,3,5,4] => 0

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [1,3,2,5,4] => 0

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [1,3,2,5,4] => 0

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [3,1,2,5,4] => 0

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [2,3,1,5,4] => 1

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [3,2,1,5,4] => 0

>>> Load all 7917 entries. <<<

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 left key permutation

Description

Return the permutation of the left key of an alternating sign matrix.
This was originally defined by Lascoux and then further studied by Aval [1].