- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤ (values match St000783The side length of the largest staircase partition fitting into a partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,.],[[.,[.,.]],.]] => [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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[[.,.],.]]]]] => [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] => 1

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

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

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

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

[.,[.,[[[.,[.,.]],.],.]]] => [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] => 2

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

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

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

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

[.,[[[.,[.,.]],.],[.,.]]] => [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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,[.,.]],[[[.,.],.],.]] => [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] => 2

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

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

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

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

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

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

>>> Load all 1522 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.

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.