Identifier
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => 1
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [2,3,1] => 0
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => 2
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 1
[[[.,.],.],.] => [1,2,3] => [1,3,2] => 2
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [2,4,3,1] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => 3
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,4,3,2] => 3
[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,4,3,2] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => [2,4,1,3] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,4,3,2] => 3
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,4,3] => 2
[[[[.,.],.],.],.] => [1,2,3,4] => [1,4,3,2] => 3
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [3,5,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,5,4,2,1] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [2,5,4,3,1] => 0
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [2,5,4,3,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,5,2,4,1] => 0
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [2,5,4,3,1] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [3,2,5,4,1] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [2,5,4,3,1] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => 4
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,5,4,3,2] => 4
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,5,4,3,2] => 4
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,5,4,3,2] => 4
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,5,4,3,2] => 4
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,1,5,4,3] => 3
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,1,5,4,3] => 3
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,5,4,3,2] => 4
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,5,4,3,2] => 4
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,5,4,3,2] => 4
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,1,5,4,3] => 3
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,5,4,3,2] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [3,5,2,1,4] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [2,5,4,1,3] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,2,5,1,4] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [2,5,4,1,3] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,5,4,3,2] => 4
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,5,4,3,2] => 4
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [2,1,5,4,3] => 3
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,5,4,3,2] => 4
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,5,4] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,5,4,3,2] => 4
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,5,4,3] => 3
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,5,4,3,2] => 4
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [5,6,4,3,2,1] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [4,6,5,3,2,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [5,4,6,3,2,1] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,6,5,3,2,1] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [3,6,5,4,2,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [3,6,5,4,2,1] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [4,3,6,5,2,1] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [3,6,5,4,2,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [5,4,3,6,2,1] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [4,6,3,5,2,1] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [3,6,5,4,2,1] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [4,3,6,5,2,1] => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [3,6,5,4,2,1] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [2,6,5,4,3,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [2,6,5,4,3,1] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [2,6,5,4,3,1] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [2,6,5,4,3,1] => 0
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [2,6,5,4,3,1] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [3,2,6,5,4,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [3,2,6,5,4,1] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [2,6,5,4,3,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [2,6,5,4,3,1] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [4,3,2,6,5,1] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [3,6,2,5,4,1] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [2,6,5,4,3,1] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [3,2,6,5,4,1] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [2,6,5,4,3,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [5,4,3,2,6,1] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [4,6,3,2,5,1] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [3,6,5,2,4,1] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [4,3,6,2,5,1] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,6,5,2,4,1] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [2,6,5,4,3,1] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [2,6,5,4,3,1] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [3,2,6,5,4,1] => 0
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [2,6,5,4,3,1] => 0
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [4,3,2,6,5,1] => 0
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [3,6,2,5,4,1] => 0
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => [2,6,5,4,3,1] => 0
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [3,2,6,5,4,1] => 0
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [2,6,5,4,3,1] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 5
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => [1,6,5,4,3,2] => 5
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => [1,6,5,4,3,2] => 5
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [1,6,5,4,3,2] => 5
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => [1,6,5,4,3,2] => 5
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => [1,6,5,4,3,2] => 5
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => [1,6,5,4,3,2] => 5
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => [1,6,5,4,3,2] => 5
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => [1,6,5,4,3,2] => 5
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => [1,6,5,4,3,2] => 5
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => [1,6,5,4,3,2] => 5
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => [1,6,5,4,3,2] => 5
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => [1,6,5,4,3,2] => 5
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => [1,6,5,4,3,2] => 5
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => [2,1,6,5,4,3] => 4
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 4
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => [2,1,6,5,4,3] => 4
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [2,1,6,5,4,3] => 4
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => [2,1,6,5,4,3] => 4
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => [1,6,5,4,3,2] => 5
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => [1,6,5,4,3,2] => 5
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => [1,6,5,4,3,2] => 5
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => [1,6,5,4,3,2] => 5
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => [1,6,5,4,3,2] => 5
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 3
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => [3,2,1,6,5,4] => 3
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => [2,6,1,5,4,3] => 3
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => [2,6,1,5,4,3] => 3
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => [1,6,5,4,3,2] => 5
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => [1,6,5,4,3,2] => 5
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => [2,1,6,5,4,3] => 4
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => [2,1,6,5,4,3] => 4
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => [1,6,5,4,3,2] => 5
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => [1,6,5,4,3,2] => 5
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => 2
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => [3,6,2,1,5,4] => 2
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => [2,6,5,1,4,3] => 2
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => [3,2,6,1,5,4] => 2
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => [2,6,5,1,4,3] => 2
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => [1,6,5,4,3,2] => 5
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => [1,6,5,4,3,2] => 5
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => [2,1,6,5,4,3] => 4
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => [1,6,5,4,3,2] => 5
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => [3,2,1,6,5,4] => 3
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => [2,6,1,5,4,3] => 3
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => [1,6,5,4,3,2] => 5
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => [2,1,6,5,4,3] => 4
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => [1,6,5,4,3,2] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [4,6,3,2,1,5] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => [3,6,5,2,1,4] => 1
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [4,3,6,2,1,5] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [3,6,5,2,1,4] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => [2,6,5,4,1,3] => 1
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => [2,6,5,4,1,3] => 1
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => [3,2,6,5,1,4] => 1
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => [2,6,5,4,1,3] => 1
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [4,3,2,6,1,5] => 1
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [3,6,2,5,1,4] => 1
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => [2,6,5,4,1,3] => 1
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [3,2,6,5,1,4] => 1
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [2,6,5,4,1,3] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => [1,6,5,4,3,2] => 5
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => [1,6,5,4,3,2] => 5
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => [1,6,5,4,3,2] => 5
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => [1,6,5,4,3,2] => 5
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => [1,6,5,4,3,2] => 5
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => [2,1,6,5,4,3] => 4
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => [2,1,6,5,4,3] => 4
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => [1,6,5,4,3,2] => 5
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => [1,6,5,4,3,2] => 5
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [3,2,1,6,5,4] => 3
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => [2,6,1,5,4,3] => 3
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => 5
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => [2,1,6,5,4,3] => 4
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => [1,6,5,4,3,2] => 5
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [4,3,2,1,6,5] => 2
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [3,6,2,1,5,4] => 2
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => [2,6,5,1,4,3] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [3,2,6,1,5,4] => 2
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [2,6,5,1,4,3] => 2
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => [1,6,5,4,3,2] => 5
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => [1,6,5,4,3,2] => 5
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => [2,1,6,5,4,3] => 4
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => [1,6,5,4,3,2] => 5
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [3,2,1,6,5,4] => 3
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [2,6,1,5,4,3] => 3
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => [1,6,5,4,3,2] => 5
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [2,1,6,5,4,3] => 4
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,6,5,4,3,2] => 5
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Description
The "bounce" of a permutation.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.