- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 11430 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 addable cells of the Ferrers diagram of an integer partition.

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.

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.