Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 2
[[.,.],.] => [1,2] => 2
[.,[.,[.,.]]] => [3,2,1] => 3
[.,[[.,.],.]] => [2,3,1] => 3
[[.,.],[.,.]] => [3,1,2] => 2
[[.,[.,.]],.] => [2,1,3] => 3
[[[.,.],.],.] => [1,2,3] => 3
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 4
[.,[.,[[.,.],.]]] => [3,4,2,1] => 4
[.,[[.,.],[.,.]]] => [4,2,3,1] => 3
[.,[[.,[.,.]],.]] => [3,2,4,1] => 4
[.,[[[.,.],.],.]] => [2,3,4,1] => 4
[[.,.],[.,[.,.]]] => [4,3,1,2] => 3
[[.,.],[[.,.],.]] => [3,4,1,2] => 3
[[.,[.,.]],[.,.]] => [4,2,1,3] => 3
[[[.,.],.],[.,.]] => [4,1,2,3] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => 4
[[.,[[.,.],.]],.] => [2,3,1,4] => 4
[[[.,.],[.,.]],.] => [3,1,2,4] => 3
[[[.,[.,.]],.],.] => [2,1,3,4] => 4
[[[[.,.],.],.],.] => [1,2,3,4] => 4
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 5
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => 4
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 5
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 5
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 4
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => 4
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 4
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 4
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 5
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 5
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 4
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 5
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 5
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 4
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => 4
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 3
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 4
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 4
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 4
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => 4
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 3
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 4
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => 4
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 3
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 4
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 5
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 5
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => 4
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 5
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 5
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 4
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => 4
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 4
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 4
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 5
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 5
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 4
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 5
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 6
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 6
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => 5
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 6
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 6
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => 5
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => 5
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => 5
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => 5
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 6
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 6
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => 5
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 6
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 6
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 5
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => 5
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => 4
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => 5
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => 5
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 5
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => 5
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 4
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => 4
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 5
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 5
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 4
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => 5
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 5
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 6
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 6
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 5
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 6
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 6
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 5
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 5
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => 5
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 5
>>> Load all 197 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 6
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 6
[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => 5
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 6
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 6
[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => 5
[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => 5
[[.,.],[.,[[.,.],[.,.]]]] => [6,4,5,3,1,2] => 4
[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => 5
[[.,.],[.,[[[.,.],.],.]]] => [4,5,6,3,1,2] => 5
[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => 4
[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => 4
[[.,.],[[.,[.,.]],[.,.]]] => [6,4,3,5,1,2] => 4
[[.,.],[[[.,.],.],[.,.]]] => [6,3,4,5,1,2] => 4
[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => 5
[[.,.],[[.,[[.,.],.]],.]] => [4,5,3,6,1,2] => 5
[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => 4
[[.,.],[[[.,[.,.]],.],.]] => [4,3,5,6,1,2] => 5
[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => 5
[[.,[.,.]],[.,[.,[.,.]]]] => [6,5,4,2,1,3] => 5
[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => 5
[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => 4
[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => 5
[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => 5
[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => 4
[[[.,.],.],[.,[[.,.],.]]] => [5,6,4,1,2,3] => 4
[[[.,.],.],[[.,.],[.,.]]] => [6,4,5,1,2,3] => 3
[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => 4
[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => 4
[[.,[.,[.,.]]],[.,[.,.]]] => [6,5,3,2,1,4] => 5
[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => 5
[[.,[[.,.],.]],[.,[.,.]]] => [6,5,2,3,1,4] => 4
[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => 4
[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => 4
[[[.,.],[.,.]],[[.,.],.]] => [5,6,3,1,2,4] => 4
[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => 4
[[[.,[.,.]],.],[[.,.],.]] => [5,6,2,1,3,4] => 4
[[[[.,.],.],.],[.,[.,.]]] => [6,5,1,2,3,4] => 4
[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => 5
[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => 5
[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => 4
[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => 5
[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => 5
[[[.,.],[.,[.,.]]],[.,.]] => [6,4,3,1,2,5] => 4
[[[.,.],[[.,.],.]],[.,.]] => [6,3,4,1,2,5] => 4
[[[.,[.,.]],[.,.]],[.,.]] => [6,4,2,1,3,5] => 4
[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => 4
[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => 5
[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => 5
[[[[.,.],[.,.]],.],[.,.]] => [6,3,1,2,4,5] => 4
[[[[.,[.,.]],.],.],[.,.]] => [6,2,1,3,4,5] => 5
[[[[[.,.],.],.],.],[.,.]] => [6,1,2,3,4,5] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 6
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 6
[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => 5
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 6
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 6
[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => 5
[[.,[[.,.],[[.,.],.]]],.] => [4,5,2,3,1,6] => 5
[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => 5
[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => 5
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 6
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 6
[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => 5
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 6
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 6
[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => 5
[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => 5
[[[.,.],[[.,.],[.,.]]],.] => [5,3,4,1,2,6] => 4
[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => 5
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => 5
[[[.,[.,.]],[.,[.,.]]],.] => [5,4,2,1,3,6] => 5
[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => 5
[[[[.,.],.],[.,[.,.]]],.] => [5,4,1,2,3,6] => 4
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => 4
[[[.,[.,[.,.]]],[.,.]],.] => [5,3,2,1,4,6] => 5
[[[.,[[.,.],.]],[.,.]],.] => [5,2,3,1,4,6] => 5
[[[[.,.],[.,.]],[.,.]],.] => [5,3,1,2,4,6] => 4
[[[[.,[.,.]],.],[.,.]],.] => [5,2,1,3,4,6] => 5
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => 5
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 6
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 6
[[[.,[[.,.],[.,.]]],.],.] => [4,2,3,1,5,6] => 5
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 6
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 6
[[[[.,.],[.,[.,.]]],.],.] => [4,3,1,2,5,6] => 5
[[[[.,.],[[.,.],.]],.],.] => [3,4,1,2,5,6] => 5
[[[[.,[.,.]],[.,.]],.],.] => [4,2,1,3,5,6] => 5
[[[[[.,.],.],[.,.]],.],.] => [4,1,2,3,5,6] => 5
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 6
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 6
[[[[[.,.],[.,.]],.],.],.] => [3,1,2,4,5,6] => 5
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 6
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 6
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => 7
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Description
The length of the first row of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the length of the first row of $P$ and $Q$.
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.