Identifier
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [2,3,1] => 0
[[.,.],[.,.]] => [3,1,2] => [2,3,1] => 0
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,4,2,1] => 0
[.,[[.,.],[.,.]]] => [4,2,3,1] => [3,4,2,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [3,2,4,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [2,3,4,1] => 0
[[.,.],[.,[.,.]]] => [4,3,1,2] => [3,4,2,1] => 0
[[.,.],[[.,.],.]] => [3,4,1,2] => [4,3,2,1] => 0
[[.,[.,.]],[.,.]] => [4,2,1,3] => [3,2,4,1] => 0
[[[.,.],.],[.,.]] => [4,1,2,3] => [2,3,4,1] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [2,3,1,4] => 0
[[[.,.],[.,.]],.] => [3,1,2,4] => [2,3,1,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,4,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [4,5,3,2,1] => 0
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [5,4,3,2,1] => 0
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [4,3,5,2,1] => 0
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [3,4,5,2,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [3,4,2,5,1] => 0
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [3,4,2,5,1] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [4,5,3,2,1] => 0
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [5,4,3,2,1] => 0
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [5,4,3,2,1] => 0
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [5,3,4,2,1] => 1
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [3,5,4,2,1] => 0
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [4,3,5,2,1] => 0
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [3,4,5,2,1] => 0
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [3,4,5,2,1] => 0
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [4,3,5,2,1] => 0
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [4,3,2,5,1] => 0
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [3,4,2,5,1] => 0
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [3,4,2,5,1] => 0
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [3,2,4,5,1] => 0
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [2,3,4,5,1] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [3,4,2,1,5] => 0
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [3,4,2,1,5] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,2,4,1,5] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [2,3,4,1,5] => 0
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [3,4,2,1,5] => 0
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [4,3,2,1,5] => 0
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [3,2,4,1,5] => 0
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [2,3,4,1,5] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,3,1,4,5] => 0
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [2,3,1,4,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [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
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [5,6,4,3,2,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [5,4,6,3,2,1] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [5,6,4,3,2,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [6,5,4,3,2,1] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [5,4,6,3,2,1] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [4,5,6,3,2,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [5,4,3,6,2,1] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [4,5,3,6,2,1] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [4,5,3,6,2,1] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [4,3,5,6,2,1] => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [3,4,5,6,2,1] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [5,6,4,3,2,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [6,5,4,3,2,1] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [6,5,4,3,2,1] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [6,4,5,3,2,1] => 1
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [4,6,5,3,2,1] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [5,4,6,3,2,1] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [4,5,6,3,2,1] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [4,5,6,3,2,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [5,4,6,3,2,1] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [5,4,3,6,2,1] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [4,5,3,6,2,1] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [4,5,3,6,2,1] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [4,3,5,6,2,1] => 0
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [3,4,5,6,2,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [5,4,3,2,6,1] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [4,5,3,2,6,1] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [4,5,3,2,6,1] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,4,5,2,6,1] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [4,5,3,2,6,1] => 0
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [5,4,3,2,6,1] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [4,3,5,2,6,1] => 0
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [3,4,5,2,6,1] => 0
>>> Load all 197 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [4,3,2,5,6,1] => 0
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => [3,4,2,5,6,1] => 0
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => [3,4,2,5,6,1] => 0
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => [3,2,4,5,6,1] => 0
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => 0
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 0
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => [6,5,4,3,2,1] => 0
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => [6,5,4,3,2,1] => 0
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => [6,4,5,3,2,1] => 1
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => [4,6,5,3,2,1] => 0
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => [6,5,4,3,2,1] => 0
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => [5,6,4,3,2,1] => 0
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => [6,4,5,3,2,1] => 1
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => [4,6,5,3,2,1] => 0
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => [6,4,3,5,2,1] => 1
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => [4,6,3,5,2,1] => 1
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => [4,6,3,5,2,1] => 1
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => [4,3,6,5,2,1] => 0
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => [3,4,6,5,2,1] => 0
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 0
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => [4,5,6,3,2,1] => 0
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => [4,5,6,3,2,1] => 0
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => [4,6,5,3,2,1] => 0
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => [6,4,5,3,2,1] => 1
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => 0
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => [5,4,6,3,2,1] => 0
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => [5,4,6,3,2,1] => 0
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => [5,6,4,3,2,1] => 0
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => [6,5,4,3,2,1] => 0
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 0
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => [4,5,3,6,2,1] => 0
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => [4,5,3,6,2,1] => 0
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => [5,4,3,6,2,1] => 0
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => [4,5,3,6,2,1] => 0
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => [5,4,3,6,2,1] => 0
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => [4,3,5,6,2,1] => 0
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => [3,4,5,6,2,1] => 0
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => [3,4,5,6,2,1] => 0
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => [4,3,5,6,2,1] => 0
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 0
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => [4,5,3,2,6,1] => 0
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => [4,5,3,2,6,1] => 0
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => [4,3,5,2,6,1] => 0
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => [3,4,5,2,6,1] => 0
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => [4,5,3,2,6,1] => 0
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => [5,4,3,2,6,1] => 0
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => [4,3,5,2,6,1] => 0
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => 0
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => [4,3,2,5,6,1] => 0
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => [3,4,2,5,6,1] => 0
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 0
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => [3,2,4,5,6,1] => 0
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 0
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [4,5,3,2,1,6] => 0
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => [4,5,3,2,1,6] => 0
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [4,3,5,2,1,6] => 0
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [3,4,5,2,1,6] => 0
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => [4,5,3,2,1,6] => 0
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => [5,4,3,2,1,6] => 0
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => [4,3,5,2,1,6] => 0
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => [3,4,5,2,1,6] => 0
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [4,3,2,5,1,6] => 0
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => [3,4,2,5,1,6] => 0
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => [3,4,2,5,1,6] => 0
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [3,2,4,5,1,6] => 0
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => [2,3,4,5,1,6] => 0
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => [4,5,3,2,1,6] => 0
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => [5,4,3,2,1,6] => 0
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => [5,4,3,2,1,6] => 0
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => [5,3,4,2,1,6] => 1
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => [3,5,4,2,1,6] => 0
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => [4,3,5,2,1,6] => 0
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => [3,4,5,2,1,6] => 0
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => [3,4,5,2,1,6] => 0
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => [4,3,5,2,1,6] => 0
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => [4,3,2,5,1,6] => 0
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => [3,4,2,5,1,6] => 0
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => [3,4,2,5,1,6] => 0
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => [3,2,4,5,1,6] => 0
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => [2,3,4,5,1,6] => 0
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 0
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => [3,4,2,1,5,6] => 0
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => [3,4,2,1,5,6] => 0
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => [3,2,4,1,5,6] => 0
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => [2,3,4,1,5,6] => 0
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => [3,4,2,1,5,6] => 0
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => [4,3,2,1,5,6] => 0
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => [3,2,4,1,5,6] => 0
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => [2,3,4,1,5,6] => 0
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 0
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => [2,3,1,4,5,6] => 0
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => [2,3,1,4,5,6] => 0
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Map
catalanization
Description
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
Map
to 132-avoiding permutation
Description
Return a 132-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 maximal element of the Sylvester class.