- FindStat

Identifier

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000913: Integer partitions ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 34337 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of ways to refine the partition into singletons.
For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$.
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.

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.