- FindStat

Identifier

Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000716: Integer partitions ⟶ ℤ

Values

[[1,4],[2],[3]] => [[1,2],[3],[4]] => [2,1,1] => [1,1] => 14

[[1],[2],[3],[4]] => [[1,2],[3],[4]] => [2,1,1] => [1,1] => 14

[[1,5],[2],[3]] => [[1,2],[3],[5]] => [2,1,1] => [1,1] => 14

[[1,5],[2],[4]] => [[1,2],[4],[5]] => [2,1,1] => [1,1] => 14

[[1,5],[3],[4]] => [[1,3],[4],[5]] => [2,1,1] => [1,1] => 14

[[2,5],[3],[4]] => [[2,3],[4],[5]] => [2,1,1] => [1,1] => 14

[[1],[2],[3],[5]] => [[1,2],[3],[5]] => [2,1,1] => [1,1] => 14

[[1],[2],[4],[5]] => [[1,2],[4],[5]] => [2,1,1] => [1,1] => 14

[[1],[3],[4],[5]] => [[1,3],[4],[5]] => [2,1,1] => [1,1] => 14

[[2],[3],[4],[5]] => [[2,3],[4],[5]] => [2,1,1] => [1,1] => 14

[[1,1,4],[2],[3]] => [[1,1,2],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,2,4],[2],[3]] => [[1,2,2],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,3,4],[2],[3]] => [[1,2,3],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,4,4],[2],[3]] => [[1,2,4],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,1],[2],[3],[4]] => [[1,1,2],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,2],[2],[3],[4]] => [[1,2,2],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,3],[2],[3],[4]] => [[1,2,3],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,4],[2],[3],[4]] => [[1,2,4],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,1,3,3],[2,2]] => [[1,1,2,2],[3,3]] => [4,2] => [2] => 21

[[1,1,3],[2,2],[3]] => [[1,1,2,2],[3,3]] => [4,2] => [2] => 21

[[1,1],[2,2],[3,3]] => [[1,1,2,2],[3,3]] => [4,2] => [2] => 21

[[1,6],[2],[3]] => [[1,2],[3],[6]] => [2,1,1] => [1,1] => 14

[[1,6],[2],[4]] => [[1,2],[4],[6]] => [2,1,1] => [1,1] => 14

[[1,6],[2],[5]] => [[1,2],[5],[6]] => [2,1,1] => [1,1] => 14

[[1,6],[3],[4]] => [[1,3],[4],[6]] => [2,1,1] => [1,1] => 14

[[1,6],[3],[5]] => [[1,3],[5],[6]] => [2,1,1] => [1,1] => 14

[[1,6],[4],[5]] => [[1,4],[5],[6]] => [2,1,1] => [1,1] => 14

[[2,6],[3],[4]] => [[2,3],[4],[6]] => [2,1,1] => [1,1] => 14

[[2,6],[3],[5]] => [[2,3],[5],[6]] => [2,1,1] => [1,1] => 14

[[2,6],[4],[5]] => [[2,4],[5],[6]] => [2,1,1] => [1,1] => 14

[[3,6],[4],[5]] => [[3,4],[5],[6]] => [2,1,1] => [1,1] => 14

[[1],[2],[3],[6]] => [[1,2],[3],[6]] => [2,1,1] => [1,1] => 14

[[1],[2],[4],[6]] => [[1,2],[4],[6]] => [2,1,1] => [1,1] => 14

[[1],[2],[5],[6]] => [[1,2],[5],[6]] => [2,1,1] => [1,1] => 14

[[1],[3],[4],[6]] => [[1,3],[4],[6]] => [2,1,1] => [1,1] => 14

[[1],[3],[5],[6]] => [[1,3],[5],[6]] => [2,1,1] => [1,1] => 14

[[1],[4],[5],[6]] => [[1,4],[5],[6]] => [2,1,1] => [1,1] => 14

[[2],[3],[4],[6]] => [[2,3],[4],[6]] => [2,1,1] => [1,1] => 14

[[2],[3],[5],[6]] => [[2,3],[5],[6]] => [2,1,1] => [1,1] => 14

[[2],[4],[5],[6]] => [[2,4],[5],[6]] => [2,1,1] => [1,1] => 14

[[3],[4],[5],[6]] => [[3,4],[5],[6]] => [2,1,1] => [1,1] => 14

[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => [3,2] => [2] => 21

[[1,1,5],[2],[3]] => [[1,1,2],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,1,5],[2],[4]] => [[1,1,2],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,1,5],[3],[4]] => [[1,1,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,2,5],[2],[3]] => [[1,2,2],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,2,5],[2],[4]] => [[1,2,2],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3,5],[2],[3]] => [[1,2,3],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,2,5],[3],[4]] => [[1,2,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,4,5],[2],[3]] => [[1,2,5],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,5,5],[2],[3]] => [[1,2,5],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,4,5],[2],[4]] => [[1,2,4],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,5,5],[2],[4]] => [[1,2,5],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3,5],[3],[4]] => [[1,3,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,4,5],[3],[4]] => [[1,3,4],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,5,5],[3],[4]] => [[1,3,5],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,2,5],[3],[4]] => [[2,2,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,3,5],[3],[4]] => [[2,3,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,4,5],[3],[4]] => [[2,3,4],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,5,5],[3],[4]] => [[2,3,5],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3],[2,4],[5]] => [[1,2,4],[3,5]] => [3,2] => [2] => 21

[[1,4],[2,5],[3]] => [[1,2,5],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,1],[2],[3],[5]] => [[1,1,2],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,1],[2],[4],[5]] => [[1,1,2],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,1],[3],[4],[5]] => [[1,1,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,2],[2],[3],[5]] => [[1,2,2],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,2],[2],[4],[5]] => [[1,2,2],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3],[2],[3],[5]] => [[1,2,3],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,2],[3],[4],[5]] => [[1,2,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3],[2],[4],[5]] => [[1,2,4],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,4],[2],[3],[5]] => [[1,2,5],[3],[4]] => [3,1,1] => [1,1] => 14

[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => [2,1,1,1] => [1,1,1] => 14

[[1,5],[2],[3],[5]] => [[1,2,5],[3],[5]] => [3,1,1] => [1,1] => 14

[[1,4],[2],[4],[5]] => [[1,2,4],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,5],[2],[4],[5]] => [[1,2,5],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,3],[3],[4],[5]] => [[1,3,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,4],[3],[4],[5]] => [[1,3,4],[4],[5]] => [3,1,1] => [1,1] => 14

[[1,5],[3],[4],[5]] => [[1,3,5],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,2],[3],[4],[5]] => [[2,2,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,3],[3],[4],[5]] => [[2,3,3],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,4],[3],[4],[5]] => [[2,3,4],[4],[5]] => [3,1,1] => [1,1] => 14

[[2,5],[3],[4],[5]] => [[2,3,5],[4],[5]] => [3,1,1] => [1,1] => 14

[[1],[2],[3],[4],[5]] => [[1,2],[3],[4],[5]] => [2,1,1,1] => [1,1,1] => 14

[[1,1,3,4],[2,2]] => [[1,1,2,2],[3,4]] => [4,2] => [2] => 21

[[1,1,4,4],[2,2]] => [[1,1,2,2],[4,4]] => [4,2] => [2] => 21

[[1,1,3,4],[2,3]] => [[1,1,2,3],[3,4]] => [4,2] => [2] => 21

[[1,1,4,4],[2,3]] => [[1,1,2,3],[4,4]] => [4,2] => [2] => 21

[[1,1,4,4],[3,3]] => [[1,1,3,3],[4,4]] => [4,2] => [2] => 21

[[1,2,3,4],[2,3]] => [[1,2,2,3],[3,4]] => [4,2] => [2] => 21

[[1,2,4,4],[2,3]] => [[1,2,2,3],[4,4]] => [4,2] => [2] => 21

[[1,2,4,4],[3,3]] => [[1,2,3,3],[4,4]] => [4,2] => [2] => 21

[[2,2,4,4],[3,3]] => [[2,2,3,3],[4,4]] => [4,2] => [2] => 21

[[1,1,1,4],[2],[3]] => [[1,1,1,2],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,1,2,4],[2],[3]] => [[1,1,2,2],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,1,3,4],[2],[3]] => [[1,1,2,3],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,1,4,4],[2],[3]] => [[1,1,2,4],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,2,2,4],[2],[3]] => [[1,2,2,2],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,2,3,4],[2],[3]] => [[1,2,2,3],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,2,4,4],[2],[3]] => [[1,2,2,4],[3],[4]] => [4,1,1] => [1,1] => 14

[[1,3,3,4],[2],[3]] => [[1,2,3,3],[3],[4]] => [4,1,1] => [1,1] => 14

>>> Load all 247 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
Consider the symplectic group

$Sp(2n)$ . Then the integer partition

$(\mu_1,\dots,\mu_k)$ of length at most

$n$ corresponds to the weight vector

$(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$ .
For example, the integer partition

$(2)$ labels the symmetric square of the vector representation, whereas the integer partition

$(1,1)$ labels the second fundamental representation.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

catabolism

Description

Remove the first row of the semistandard tableau and insert it back using column Schensted insertion, starting with the largest number.
The algorithm for column-inserting an entry

$k$ into tableau

$T$ is as follows:
If

$k$ is larger than all entries in the first column, place

$k$ at the bottom of the first column and the procedure is finished. Otherwise, place

$k$ in the first column, replacing the smallest entry,

$y$ , greater or equal to than

$k$ . Now insert

$y$ into the second column using the same procedure: if

$y$ is greater than all entries in the second column, place it at the bottom of that column (provided that the result is still a tableau). Otherwise, place

$y$ in the second column, replacing, or 'bumping', the smallest entry,

$z$ , larger than or equal to

$y$ . Continue the procedure until we have placed a bumped entry at the bottom of a column (or on its own in a new column).

Map

shape

Description

Return the shape of a tableau.