Identifier
Values
[1,1,0,0,1,0] => [2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0] => 0
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 0
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 0
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => 0
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0] => 0
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => 0
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 0
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => 0
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0] => 0
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0] => 0
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0] => 0
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0] => 0
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,6,1,4,5,2] => [1,1,1,0,1,1,1,0,0,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 0
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => 0
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0] => 0
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0] => 0
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,2,3,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0] => 0
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,6,4,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0] => 0
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => [6,2,3,4,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [5,2,3,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [5,2,3,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => [6,2,3,4,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [6,2,3,7,5,1,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [4,2,7,1,6,5,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => [6,2,4,3,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => [5,2,3,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => [5,2,3,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [5,2,7,6,1,4,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [5,2,7,6,1,4,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [5,2,7,4,1,6,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 0
[1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => [6,2,3,4,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [6,2,3,7,5,1,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [6,2,7,5,4,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [3,7,1,4,5,6,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => [3,7,1,4,5,6,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => [3,6,1,4,7,2,5] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [3,7,1,4,6,5,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [3,7,1,4,5,6,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0] => 0
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [3,5,1,7,2,6,4] => [1,1,1,0,1,1,0,0,1,1,0,0,0,0] => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [3,7,1,6,5,4,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [3,7,1,6,5,4,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [3,7,1,5,4,6,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [3,6,1,4,7,2,5] => [1,1,1,0,1,1,1,0,0,0,1,0,0,0] => 0
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [3,6,1,7,5,2,4] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [3,7,1,6,5,4,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [3,7,1,4,5,6,2] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [6,4,3,2,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [6,7,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 0
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => [6,4,3,2,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [5,3,2,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => [5,3,2,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => [6,3,2,4,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [6,3,2,7,5,1,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0] => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [4,2,7,1,6,5,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [4,2,7,1,5,6,3] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [4,7,5,1,3,6,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [4,7,5,1,3,6,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [4,7,6,1,5,3,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [4,7,6,1,5,3,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [4,7,5,1,3,6,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [4,6,3,1,7,2,5] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0] => 0
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [4,7,6,1,5,3,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [4,7,6,1,5,3,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [4,7,3,1,5,6,2] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,3,4,1,7,2,6] => [6,4,3,2,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [5,2,3,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [5,2,3,7,1,6,4] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [5,2,7,6,1,4,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [5,2,7,6,1,4,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [5,2,7,4,1,6,3] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [5,7,6,4,1,3,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [5,7,6,4,1,3,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [5,7,6,4,1,3,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [5,7,4,3,1,6,2] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [6,2,3,4,7,1,5] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [6,2,3,7,5,1,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [6,2,7,5,4,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [6,7,5,4,3,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 0
search for individual values
searching the database for the individual values of this statistic
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.