- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[.,[[.,.],.]],[.,.]]] => [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] => 0

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

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

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

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

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

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

[.,[[[.,[.,.]],[.,.]],.]] => [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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1522 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting

$S_{(6,3)}$ to

$\mathfrak S_8$ yields

$S_{(5,3)}\oplus S_{(6,2)}$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to

$\mathfrak S_7$ yields

$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$ of degrees 14, 14 and 6. However, restricting to

$\mathfrak S_6$ yields

$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition

$(6,3)$ gives 3.
This is related to

$2$ -saturations of Welter's game, see [1, Corollary 1.2].

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.