- 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
St001124: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St000318The number of addable cells of the Ferrers diagram of an integer partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1602 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than St000159The number of distinct parts of the integer partition., the number of distinct parts of the partition.

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.