- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤ

Values

[[[.,.],.],.] => [1,2,3] => [1,1,1] => [1,1] => 2

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [2,2] => [2] => 1

[.,[[.,.],[.,.]]] => [4,2,3,1] => [2,1,1] => [1,1] => 2

[[.,.],[[.,.],.]] => [3,4,1,2] => [2,2] => [2] => 1

[[.,[.,[.,.]]],.] => [3,2,1,4] => [2,1,1] => [1,1] => 2

[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,1] => [1,1] => 2

[[[[.,.],.],.],.] => [1,2,3,4] => [1,1,1,1] => [1,1,1] => 2

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [2,2,1] => [2,1] => 1

[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [3,2] => [2] => 1

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,2] => [2] => 1

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [3,2] => [2] => 1

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [2,2,1] => [2,1] => 1

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [2,1,1,1] => [1,1,1] => 2

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [3,2] => [2] => 1

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [3,1,1] => [1,1] => 2

[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [2,2,1] => [2,1] => 1

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [3,2] => [2] => 1

[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [3,2] => [2] => 1

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [3,2] => [2] => 1

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [3,2] => [2] => 1

[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [3,1,1] => [1,1] => 2

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [2,2,1] => [2,1] => 1

[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [2,1,1,1] => [1,1,1] => 2

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [3,1,1] => [1,1] => 2

[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [2,2,1] => [2,1] => 1

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [3,1,1] => [1,1] => 2

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [2,1,1,1] => [1,1,1] => 2

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [3,1,1] => [1,1] => 2

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [3,1,1] => [1,1] => 2

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,1,1] => [1,1,1] => 2

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,1,1,1,1] => [1,1,1,1] => 2

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [2,2,2] => [2,2] => 1

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [4,2] => [2] => 1

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [4,2] => [2] => 1

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,2] => [2] => 1

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [2,2,1,1] => [2,1,1] => 2

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [4,1,1] => [1,1] => 2

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [3,2,1] => [2,1] => 1

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [4,2] => [2] => 1

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [3,2,1] => [2,1] => 1

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [3,3] => [3] => 1

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [4,2] => [2] => 1

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [2,2,2] => [2,2] => 1

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [4,2] => [2] => 1

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [3,2,1] => [2,1] => 1

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [4,2] => [2] => 1

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [2,2,1,1] => [2,1,1] => 2

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [3,2,1] => [2,1] => 1

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [3,2,1] => [2,1] => 1

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [2,2,1,1] => [2,1,1] => 2

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [2,1,1,1,1] => [1,1,1,1] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [3,2,1] => [2,1] => 1

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [3,3] => [3] => 1

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,2] => [2] => 1

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [3,3] => [3] => 1

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [3,2,1] => [2,1] => 1

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [3,1,1,1] => [1,1,1] => 2

[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [4,2] => [2] => 1

[.,[[[[.,.],[.,.]],.],.]] => [4,2,3,5,6,1] => [4,1,1] => [1,1] => 2

[[.,.],[.,[.,[.,[.,.]]]]] => [6,5,4,3,1,2] => [4,2] => [2] => 1

[[.,.],[.,[.,[[.,.],.]]]] => [5,6,4,3,1,2] => [2,2,2] => [2,2] => 1

[[.,.],[.,[[.,[.,.]],.]]] => [5,4,6,3,1,2] => [4,2] => [2] => 1

[[.,.],[[.,.],[.,[.,.]]]] => [6,5,3,4,1,2] => [4,1,1] => [1,1] => 2

[[.,.],[[.,.],[[.,.],.]]] => [5,6,3,4,1,2] => [2,2,1,1] => [2,1,1] => 2

[[.,.],[[.,[.,[.,.]]],.]] => [5,4,3,6,1,2] => [3,2,1] => [2,1] => 1

[[.,.],[[[.,.],[.,.]],.]] => [5,3,4,6,1,2] => [4,2] => [2] => 1

[[.,.],[[[[.,.],.],.],.]] => [3,4,5,6,1,2] => [3,3] => [3] => 1

[[.,[.,.]],[.,[[.,.],.]]] => [5,6,4,2,1,3] => [4,2] => [2] => 1

[[.,[.,.]],[[.,.],[.,.]]] => [6,4,5,2,1,3] => [4,2] => [2] => 1

[[.,[.,.]],[[.,[.,.]],.]] => [5,4,6,2,1,3] => [2,2,2] => [2,2] => 1

[[.,[.,.]],[[[.,.],.],.]] => [4,5,6,2,1,3] => [4,2] => [2] => 1

[[[.,.],.],[.,[.,[.,.]]]] => [6,5,4,1,2,3] => [4,2] => [2] => 1

[[[.,.],.],[[.,[.,.]],.]] => [5,4,6,1,2,3] => [4,2] => [2] => 1

[[[.,.],.],[[[.,.],.],.]] => [4,5,6,1,2,3] => [2,2,2] => [2,2] => 1

[[.,[.,[.,.]]],[[.,.],.]] => [5,6,3,2,1,4] => [3,2,1] => [2,1] => 1

[[.,[[.,.],.]],[[.,.],.]] => [5,6,2,3,1,4] => [4,2] => [2] => 1

[[[.,.],[.,.]],[.,[.,.]]] => [6,5,3,1,2,4] => [3,2,1] => [2,1] => 1

[[[.,[.,.]],.],[.,[.,.]]] => [6,5,2,1,3,4] => [3,3] => [3] => 1

[[[[.,.],.],.],[[.,.],.]] => [5,6,1,2,3,4] => [3,3] => [3] => 1

[[.,[.,[.,[.,.]]]],[.,.]] => [6,4,3,2,1,5] => [3,2,1] => [2,1] => 1

[[.,[.,[[.,.],.]]],[.,.]] => [6,3,4,2,1,5] => [3,3] => [3] => 1

[[.,[[.,.],[.,.]]],[.,.]] => [6,4,2,3,1,5] => [3,3] => [3] => 1

[[.,[[.,[.,.]],.]],[.,.]] => [6,3,2,4,1,5] => [3,2,1] => [2,1] => 1

[[.,[[[.,.],.],.]],[.,.]] => [6,2,3,4,1,5] => [3,1,1,1] => [1,1,1] => 2

[[[[.,.],.],[.,.]],[.,.]] => [6,4,1,2,3,5] => [4,2] => [2] => 1

[[[.,[.,[.,.]]],.],[.,.]] => [6,3,2,1,4,5] => [4,2] => [2] => 1

[[[.,[[.,.],.]],.],[.,.]] => [6,2,3,1,4,5] => [4,1,1] => [1,1] => 2

[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [2,2,1,1] => [2,1,1] => 2

[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [4,1,1] => [1,1] => 2

[[.,[.,[[.,.],[.,.]]]],.] => [5,3,4,2,1,6] => [3,2,1] => [2,1] => 1

[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => [3,2,1] => [2,1] => 1

[[.,[[.,.],[.,[.,.]]]],.] => [5,4,2,3,1,6] => [3,2,1] => [2,1] => 1

[[.,[[.,[.,.]],[.,.]]],.] => [5,3,2,4,1,6] => [2,2,1,1] => [2,1,1] => 2

[[.,[[[.,.],.],[.,.]]],.] => [5,2,3,4,1,6] => [2,1,1,1,1] => [1,1,1,1] => 2

[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [3,2,1] => [2,1] => 1

[[.,[[[.,.],[.,.]],.]],.] => [4,2,3,5,1,6] => [3,1,1,1] => [1,1,1] => 2

[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => [4,1,1] => [1,1] => 2

[[[.,.],[.,[.,[.,.]]]],.] => [5,4,3,1,2,6] => [4,1,1] => [1,1] => 2

[[[.,.],[.,[[.,.],.]]],.] => [4,5,3,1,2,6] => [2,2,1,1] => [2,1,1] => 2

[[[.,.],[[.,[.,.]],.]],.] => [4,3,5,1,2,6] => [3,2,1] => [2,1] => 1

[[[.,[.,.]],[[.,.],.]],.] => [4,5,2,1,3,6] => [3,2,1] => [2,1] => 1

>>> Load all 1347 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The smallest positive integer that does not appear twice in the partition.

Map

first row removal

Description

Removes the first entry of an integer partition

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.

Map

cycle type

Description

The cycle type of a permutation as a partition.