- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤ

Values

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,1,1,1] => [1,1,1] => 0

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [2,1,1,1] => [1,1,1] => 0

[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [2,1,1,1] => [1,1,1] => 0

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,1,1,1] => [1,1,1] => 0

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [2,1,1,1] => [1,1,1] => 0

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [2,2,1] => [2,1] => 0

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [2,1,1,1] => [1,1,1] => 0

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,1,1,1] => [1,1,1] => 0

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,2,1] => [2,1] => 0

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [2,2,1] => [2,1] => 0

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [2,1,1,1] => [1,1,1] => 0

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

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [3,1,1,1] => [1,1,1] => 0

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [3,1,1,1] => [1,1,1] => 0

[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [2,2,1,1] => [2,1,1] => 0

[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [3,1,1,1] => [1,1,1] => 0

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [3,1,1,1] => [1,1,1] => 0

[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [3,1,1,1] => [1,1,1] => 0

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [2,2,1,1] => [2,1,1] => 0

[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [3,2,1] => [2,1] => 0

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [2,2,1,1] => [2,1,1] => 0

[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [3,2,1] => [2,1] => 0

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [3,2,1] => [2,1] => 0

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [3,2,1] => [2,1] => 0

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [2,1,1,1,1] => [1,1,1,1] => 0

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [3,1,1,1] => [1,1,1] => 0

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [3,2,1] => [2,1] => 0

[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [3,1,1,1] => [1,1,1] => 0

[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => [2,1,1,1,1] => [1,1,1,1] => 0

[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => [3,1,1,1] => [1,1,1] => 0

[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => [3,1,1,1] => [1,1,1] => 0

[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [3,1,1,1] => [1,1,1] => 0

[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => [3,1,1,1] => [1,1,1] => 0

[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => [3,2,1] => [2,1] => 0

[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => [3,1,1,1] => [1,1,1] => 0

[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => [2,2,1,1] => [2,1,1] => 0

[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => [3,2,1] => [2,1] => 0

[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => [3,2,1] => [2,1] => 0

[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [3,2,1] => [2,1] => 0

[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => [3,1,1,1] => [1,1,1] => 0

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

[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => [3,2,1] => [2,1] => 0

[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => [3,2,1] => [2,1] => 0

[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => [3,2,1] => [2,1] => 0

[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => [3,2,1] => [2,1] => 0

[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => [2,2,1,1] => [2,1,1] => 0

[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => [3,2,1] => [2,1] => 0

[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => [3,2,1] => [2,1] => 0

[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => [3,2,1] => [2,1] => 0

[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => [3,2,1] => [2,1] => 0

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

[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => [3,2,1] => [2,1] => 0

[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => [2,1,1,1,1] => [1,1,1,1] => 0

[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => [3,1,1,1] => [1,1,1] => 0

[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => [3,1,1,1] => [1,1,1] => 0

[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => [3,1,1,1] => [1,1,1] => 0

[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => [3,1,1,1] => [1,1,1] => 0

[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => [3,2,1] => [2,1] => 0

[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => [3,1,1,1] => [1,1,1] => 0

[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => [3,1,1,1] => [1,1,1] => 0

[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => [3,2,1] => [2,1] => 0

[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [3,2,1] => [2,1] => 0

[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => [3,1,1,1] => [1,1,1] => 0

[.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0

[.,[.,[.,[.,[.,[[.,.],.]]]]]] => [6,7,5,4,3,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[.,[.,[.,[.,[[.,.],[.,.]]]]]] => [5,7,6,4,3,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[.,[.,[.,[.,[[.,[.,.]],.]]]]] => [6,5,7,4,3,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[.,[.,[.,[.,[[[.,.],.],.]]]]] => [5,6,7,4,3,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[.,.],[.,[.,.]]]]]] => [4,7,6,5,3,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[.,[.,[.,[[.,.],[[.,.],.]]]]] => [4,6,7,5,3,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[.,[.,.]],[.,.]]]]] => [5,4,7,6,3,2,1] => [2,2,1,1,1] => [2,1,1,1] => 0

[.,[.,[.,[[[.,.],.],[.,.]]]]] => [4,5,7,6,3,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[.,[.,[.,.]]],.]]]] => [6,5,4,7,3,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[.,[.,[.,[[.,[[.,.],.]],.]]]] => [5,6,4,7,3,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[[.,.],[.,.]],.]]]] => [4,6,5,7,3,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[[.,[.,.]],.],.]]]] => [5,4,6,7,3,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[.,[[[[.,.],.],.],.]]]] => [4,5,6,7,3,2,1] => [4,1,1,1] => [1,1,1] => 0

[.,[.,[[.,.],[.,[.,[.,.]]]]]] => [3,7,6,5,4,2,1] => [2,1,1,1,1,1] => [1,1,1,1,1] => 1

[.,[.,[[.,.],[.,[[.,.],.]]]]] => [3,6,7,5,4,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[[.,.],[[.,.],[.,.]]]]] => [3,5,7,6,4,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[[.,.],[[.,[.,.]],.]]]] => [3,6,5,7,4,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[[.,.],[[[.,.],.],.]]]] => [3,5,6,7,4,2,1] => [4,1,1,1] => [1,1,1] => 0

[.,[.,[[.,[.,.]],[.,[.,.]]]]] => [4,3,7,6,5,2,1] => [2,2,1,1,1] => [2,1,1,1] => 0

[.,[.,[[.,[.,.]],[[.,.],.]]]] => [4,3,6,7,5,2,1] => [3,2,1,1] => [2,1,1] => 0

[.,[.,[[[.,.],.],[.,[.,.]]]]] => [3,4,7,6,5,2,1] => [3,1,1,1,1] => [1,1,1,1] => 0

[.,[.,[[[.,.],.],[[.,.],.]]]] => [3,4,6,7,5,2,1] => [4,1,1,1] => [1,1,1] => 0

>>> Load all 1226 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Map

Robinson-Schensted tableau shape

Description

Sends a permutation to its Robinson-Schensted tableau shape.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to the shape of its corresponding insertion and recording tableau.

Map

first row removal

Description

Removes the first entry of an integer partition

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.