Identifier
Values
[1] => [1] => [1] => [1,0] => 3
[1,2] => [1,2] => [1,2] => [1,0,1,0] => 4
[2,1] => [1,2] => [1,2] => [1,0,1,0] => 4
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 5
[1,3,2] => [1,3,2] => [1,2,3] => [1,0,1,0,1,0] => 5
[2,1,3] => [1,3,2] => [1,2,3] => [1,0,1,0,1,0] => 5
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 5
[3,1,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 5
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 5
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 7
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 7
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 7
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 7
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 7
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => 7
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[1,4,2,5,3] => [1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 8
[1,4,3,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 8
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[1,4,5,2,3] => [1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[1,4,5,3,2] => [1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,5,2,4,3] => [1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,5,3,2,4] => [1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,1,3,4,5] => [1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,1,3,5,4] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[2,1,4,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[2,1,4,5,3] => [1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,3,1,4,5] => [1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,3,5,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,4,1,3,5] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[2,4,1,5,3] => [1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,4,3,1,5] => [1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 10
[2,4,3,5,1] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,4,5,1,3] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,4,5,3,1] => [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,5,1,3,4] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,5,1,4,3] => [1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 8
[2,5,3,1,4] => [1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 8
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[2,5,4,1,3] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[3,1,2,4,5] => [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,1,2,5,4] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[3,1,4,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 8
[3,1,4,5,2] => [1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[3,1,5,2,4] => [1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 10
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[3,2,1,4,5] => [1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[3,2,4,1,5] => [1,5,2,4,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => 10
[3,2,4,5,1] => [1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,2,5,1,4] => [1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => 8
[3,2,5,4,1] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[3,4,1,2,5] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[3,4,2,5,1] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,5,1,2,4] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,5,1,4,2] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
>>> Load all 153 entries. <<<
[3,5,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[3,5,2,4,1] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,5,4,1,2] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,1,2,5,3] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[4,1,3,2,5] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,1,3,5,2] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[4,1,5,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[4,2,1,3,5] => [1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[4,2,1,5,3] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[4,2,3,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,2,5,1,3] => [1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,2,5,3,1] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[4,3,1,2,5] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[4,3,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[4,3,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => 10
[4,3,2,5,1] => [1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => 8
[4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,5,1,3,2] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,5,2,1,3] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,1,2,4,3] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,1,3,2,4] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,1,3,4,2] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,1,4,2,3] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[5,1,4,3,2] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[5,2,1,3,4] => [1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,2,1,4,3] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[5,2,3,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,2,4,1,3] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,2,4,3,1] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,3,1,2,4] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,3,1,4,2] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[5,3,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => 8
[5,3,2,4,1] => [1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,4,1,3,2] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,4,2,1,3] => [1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 7
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Description
The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.