- FindStat

Identifier

Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001697: Standard tableaux ⟶ ℤ

Values

[1] => [1] => [[1]] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 27783 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 shifted natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of shape $\lambda$, size $n$ with natural descent set $D$ is then $b(\lambda)+\sum_{d\in D} n-d$, where $b(\lambda) = \sum_i (i-1)\lambda_i$.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.

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.