Identifier
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [1,2] => 0
[[.,.],.] => [1,2] => [2,1] => 0
[.,[.,[.,.]]] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],.]] => [2,3,1] => [2,1,3] => 0
[[.,.],[.,.]] => [1,3,2] => [2,3,1] => 0
[[.,[.,.]],.] => [2,1,3] => [3,1,2] => 0
[[[.,.],.],.] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,1,2,4] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [2,1,3,4] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,3,1,4] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [3,2,1,4] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [2,3,4,1] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [3,2,4,1] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => [3,4,1,2] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [3,4,2,1] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [4,1,2,3] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [4,2,1,3] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => [4,2,3,1] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [4,3,1,2] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,1,2,3,5] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [3,1,2,4,5] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [3,4,1,2,5] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [4,3,1,2,5] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [2,1,3,4,5] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [4,2,1,3,5] => 0
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [2,3,1,4,5] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [3,2,1,4,5] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,2,4,1,5] => 0
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [3,4,2,1,5] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [4,2,3,1,5] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [4,2,3,5,1] => 1
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [3,2,4,5,1] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [3,4,2,5,1] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [4,3,2,5,1] => 0
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [4,3,5,1,2] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [4,3,5,2,1] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [4,5,2,1,3] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [5,1,2,3,4] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [5,3,1,2,4] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [5,2,1,3,4] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [5,2,3,1,4] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [5,3,2,1,4] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [5,3,2,4,1] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [5,3,4,1,2] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [5,4,1,2,3] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [5,4,2,1,3] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [5,4,3,1,2] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [5,1,2,3,4,6] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [4,1,2,3,5,6] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [4,5,1,2,3,6] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [5,4,1,2,3,6] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [3,1,2,4,5,6] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [5,3,1,2,4,6] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [3,4,1,2,5,6] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [4,3,1,2,5,6] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [3,4,5,1,2,6] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [4,3,5,1,2,6] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [4,5,3,1,2,6] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [5,3,4,1,2,6] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [5,4,3,1,2,6] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [2,1,3,4,5,6] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [5,2,1,3,4,6] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [4,2,1,3,5,6] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [4,5,2,1,3,6] => 0
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [5,4,2,1,3,6] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [2,3,1,4,5,6] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [5,2,3,1,4,6] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [3,2,1,4,5,6] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [5,3,2,1,4,6] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [2,3,4,1,5,6] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [3,2,4,1,5,6] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [3,4,2,1,5,6] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [4,2,3,1,5,6] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [4,3,2,1,5,6] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [2,3,4,5,1,6] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [4,2,3,5,1,6] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [3,2,4,5,1,6] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [3,4,2,5,1,6] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,3,2,5,1,6] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [3,4,5,2,1,6] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [4,3,5,2,1,6] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [4,5,2,3,1,6] => 0
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [4,5,3,2,1,6] => 0
>>> Load all 197 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [5,2,3,4,1,6] => 0
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [5,3,2,4,1,6] => 0
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => [5,3,4,2,1,6] => 0
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [5,4,2,3,1,6] => 0
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [5,4,3,2,1,6] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => 0
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => [5,2,3,4,6,1] => 1
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => [4,2,3,5,6,1] => 1
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [4,5,2,3,6,1] => 1
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => [5,4,2,3,6,1] => 1
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => [3,2,4,5,6,1] => 0
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => [5,3,2,4,6,1] => 0
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => [3,4,2,5,6,1] => 0
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => [4,3,2,5,6,1] => 0
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => [3,4,5,2,6,1] => 0
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => [4,3,5,2,6,1] => 0
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => [4,5,3,2,6,1] => 0
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => [5,3,4,2,6,1] => 0
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => [5,4,3,2,6,1] => 0
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => [3,4,5,6,1,2] => 0
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => [5,3,4,6,1,2] => 1
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => [4,3,5,6,1,2] => 0
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [4,5,3,6,1,2] => 0
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => [5,4,3,6,1,2] => 0
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => 0
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => [5,3,4,6,2,1] => 1
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => [4,3,5,6,2,1] => 0
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => [4,5,3,6,2,1] => 0
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => [5,4,3,6,2,1] => 0
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => [4,5,6,1,2,3] => 0
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => [5,4,6,1,2,3] => 0
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => [4,5,6,2,1,3] => 0
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => [5,4,6,2,1,3] => 0
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => 0
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => [5,4,6,2,3,1] => 0
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => [4,5,6,3,1,2] => 0
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => [5,4,6,3,1,2] => 0
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => 0
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => [5,4,6,3,2,1] => 0
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => [5,6,1,2,3,4] => 0
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => [5,6,3,1,2,4] => 1
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => [5,6,2,1,3,4] => 0
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => [5,6,2,3,1,4] => 0
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => [5,6,3,2,1,4] => 0
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => [5,6,2,3,4,1] => 0
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => [5,6,3,2,4,1] => 0
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 0
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => 0
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => [5,6,4,1,2,3] => 0
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => [5,6,4,2,1,3] => 0
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => [5,6,4,2,3,1] => 0
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => [5,6,4,3,1,2] => 0
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 0
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [6,1,2,3,4,5] => 0
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [6,4,1,2,3,5] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => [6,3,1,2,4,5] => 1
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [6,3,4,1,2,5] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [6,4,3,1,2,5] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => [6,2,1,3,4,5] => 0
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => [6,4,2,1,3,5] => 0
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => [6,2,3,1,4,5] => 0
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => [6,3,2,1,4,5] => 0
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [6,2,3,4,1,5] => 0
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [6,3,2,4,1,5] => 0
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => [6,3,4,2,1,5] => 0
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [6,4,2,3,1,5] => 0
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [6,4,3,2,1,5] => 0
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => [6,2,3,4,5,1] => 0
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => [6,4,2,3,5,1] => 1
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => [6,3,2,4,5,1] => 0
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => [6,3,4,2,5,1] => 0
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => [6,4,3,2,5,1] => 0
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => [6,3,4,5,1,2] => 0
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => [6,4,3,5,1,2] => 0
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => 0
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => [6,4,3,5,2,1] => 0
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [6,4,5,1,2,3] => 0
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => [6,4,5,2,1,3] => 0
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => [6,4,5,3,1,2] => 0
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [6,5,1,2,3,4] => 0
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [6,5,3,1,2,4] => 1
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => [6,5,2,1,3,4] => 0
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [6,5,2,3,1,4] => 0
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [6,5,3,2,1,4] => 0
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => [6,5,2,3,4,1] => 0
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => [6,5,3,2,4,1] => 0
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => [6,5,3,4,1,2] => 0
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [6,5,4,1,2,3] => 0
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [6,5,4,2,1,3] => 0
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => 0
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [6,5,4,3,1,2] => 0
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
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Description
The number of right ropes of a permutation.
Let $\pi$ be a permutation of length $n$. A raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents, St000441The number of successions of a permutation., and a right rope is a large ascent after a raft of $\pi$.
See Definition 3.10 and Example 3.11 in [1].
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.
Map
weak order rowmotion
Description
Return the reversal of the permutation obtained by inverting the corresponding Laguerre heap.
This map is the composite of Mp00241invert Laguerre heap and Mp00064reverse.
Conjecturally, it is also the rowmotion on the weak order:
Any semidistributive lattice $L$ has a canonical labeling of the edges of its Hasse diagram by its join irreducible elements (see [1] and [2]). Rowmotion on this lattice is the bijection which takes an element $x \in L$ with a given set of down-labels to the unique element $y \in L$ which has that set as its up-labels (see [2] and [3]). For example, if the lattice is the distributive lattice $J(P)$ of order ideals of a finite poset $P$, then this reduces to ordinary rowmotion on the order ideals of $P$.
The weak order (a.k.a. permutohedral order) on the permutations in $S_n$ is a semidistributive lattice. In this way, we obtain an action of rowmotion on the set of permutations in $S_n$.
Note that the dynamics of weak order rowmotion is poorly understood. A collection of nontrivial homomesies is described in Corollary 6.14 of [4].