Identifier
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [1,2] => 0
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],.]] => [2,3,1] => [1,2,3] => 0
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => 1
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,2,3,4] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,2,4,3] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,2,4,3] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,2,3] => 1
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,.]] => [2,1,4,3] => [1,4,2,3] => 1
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,2,3] => 1
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,4,2,3] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,4,2] => 1
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,2,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,2,3,5,4] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,3,5,4] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,3,4,5] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,2,5,3,4] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,2,4,5,3] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,2,5,3,4] => 1
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,2,3,5,4] => 1
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,2,5,3,4] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,2,5,3,4] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,2,4,3,5] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,2,4,5,3] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,4,5,2,3] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,3,5,2,4] => 1
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,4,2,3,5] => 1
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [1,5,2,3,4] => 1
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [1,4,5,2,3] => 2
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [1,5,2,3,4] => 1
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [1,5,2,3,4] => 1
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [1,3,5,2,4] => 1
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,2,3,4] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,5,2,3,4] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,5,2,4,3] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,5,2,4,3] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,5,2,3,4] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,2,5,3] => 1
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,4,2,3,5] => 1
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,5,2,3] => 2
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,4,5,2,3] => 2
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,4,5,2] => 1
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,2,3,4,5,6] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [1,2,3,4,6,5] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,3,4,6,5] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,3,4,5,6] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [1,2,3,6,4,5] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [1,2,3,5,6,4] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [1,2,3,6,4,5] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [1,2,3,4,6,5] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,2,3,6,4,5] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,2,3,6,4,5] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [1,2,3,5,4,6] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,2,3,5,6,4] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,3,4,5,6] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [1,2,6,3,4,5] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [1,2,5,6,3,4] => 2
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [1,2,4,6,3,5] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [1,2,5,3,4,6] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [1,2,4,5,6,3] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [1,2,6,3,4,5] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [1,2,5,6,3,4] => 2
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [1,2,3,6,4,5] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [1,2,3,5,6,4] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [1,2,6,3,4,5] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [1,2,6,3,4,5] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [1,2,4,3,6,5] => 2
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,2,4,6,3,5] => 1
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [1,2,3,4,6,5] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,2,6,3,4,5] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,2,6,3,4,5] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [1,2,6,3,5,4] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,2,6,3,5,4] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,2,6,3,4,5] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [1,2,5,3,6,4] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [1,2,4,5,3,6] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,2,5,3,4,6] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [1,2,3,5,4,6] => 1
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [1,2,5,6,3,4] => 2
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [1,2,5,6,3,4] => 2
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => [1,2,4,3,5,6] => 1
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [1,2,4,5,6,3] => 1
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 1
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => [1,5,6,2,3,4] => 1
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => [1,4,6,2,3,5] => 2
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [1,5,2,3,4,6] => 1
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => [1,4,5,6,2,3] => 1
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => [1,3,6,2,4,5] => 1
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => [1,3,5,6,2,4] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => [1,4,2,3,6,5] => 2
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => [1,3,4,6,2,5] => 1
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => [1,5,2,3,6,4] => 1
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => [1,4,5,2,3,6] => 2
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => [1,3,5,2,4,6] => 1
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => [1,4,2,3,5,6] => 1
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => [1,3,4,5,6,2] => 1
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => [1,6,2,3,4,5] => 1
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => [1,5,6,2,3,4] => 1
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => [1,4,6,2,3,5] => 2
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [1,5,2,3,4,6] => 1
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => [1,4,5,6,2,3] => 1
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 1
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => [1,2,5,6,3,4] => 2
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => [1,2,4,6,3,5] => 1
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => [1,2,5,3,4,6] => 1
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => [1,2,4,5,6,3] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => [1,6,2,3,4,5] => 1
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => [1,5,6,2,3,4] => 1
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => [1,6,2,3,4,5] => 1
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => [1,5,6,2,3,4] => 1
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => [1,3,2,6,4,5] => 2
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => [1,3,2,5,6,4] => 2
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => [1,3,6,2,4,5] => 1
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => [1,3,5,6,2,4] => 2
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => [1,2,3,6,4,5] => 1
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => 1
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => [1,6,2,3,4,5] => 1
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => [1,6,2,3,4,5] => 1
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => [1,6,2,4,3,5] => 1
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => [1,6,2,4,3,5] => 1
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => [1,6,2,3,4,5] => 1
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => [1,4,2,6,3,5] => 1
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => [1,3,4,2,6,5] => 2
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => [1,4,2,3,6,5] => 2
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => [1,4,6,2,3,5] => 2
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => [1,4,6,2,3,5] => 2
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => [1,3,4,6,2,5] => 1
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 1
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [1,6,2,3,4,5] => 1
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => [1,6,2,3,5,4] => 1
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [1,6,2,3,5,4] => 1
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [1,6,2,3,4,5] => 1
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => [1,6,2,5,3,4] => 1
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => [1,6,2,4,5,3] => 1
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => [1,6,2,5,3,4] => 1
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => [1,6,2,3,5,4] => 1
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [1,6,2,5,3,4] => 1
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [1,6,2,5,3,4] => 1
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => [1,6,2,4,3,5] => 1
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [1,6,2,4,5,3] => 1
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [1,6,2,3,4,5] => 1
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => [1,5,2,6,3,4] => 2
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => [1,4,5,2,6,3] => 2
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => [1,3,5,2,6,4] => 1
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => [1,4,2,6,3,5] => 1
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => [1,3,4,5,2,6] => 1
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => [1,5,2,3,6,4] => 1
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => [1,4,5,2,3,6] => 2
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => [1,2,5,3,6,4] => 1
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => 1
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [1,5,2,3,4,6] => 1
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => [1,5,2,3,4,6] => 1
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => [1,3,5,2,4,6] => 1
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [1,5,6,2,3,4] => 1
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [1,5,6,2,3,4] => 1
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => [1,5,6,2,4,3] => 2
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [1,5,6,2,4,3] => 2
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [1,5,6,2,3,4] => 1
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => [1,4,2,5,6,3] => 1
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => 1
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => [1,4,2,3,5,6] => 1
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [1,4,5,6,2,3] => 1
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [1,4,5,6,2,3] => 1
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 1
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
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Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see St000031The number of cycles in the cycle decomposition of a permutation.) and the number of fixed points of $\pi$ (see St000022The number of fixed points of a permutation.).
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
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.