Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000937: Integer partitions ⟶ ℤ
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [2] => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [2] => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 2
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [2] => 2
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [2] => 2
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [3] => 3
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [2] => 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [1,1] => 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 2
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [2] => 2
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [2] => 2
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [3] => 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 2
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,1] => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 2
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [2,2,2] => 5
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 2
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [2] => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [3,2] => 4
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [2] => 2
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [2] => 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [3,3] => 5
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [3] => 3
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [3] => 3
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [4] => 5
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1] => 1
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => [2,1,1] => [1,1] => 1
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => [2,2,1] => [2,1] => 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => [2,2] => [2] => 2
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => [3,2] => [2] => 2
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 3
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 2
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 2
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 2
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]] => [2,1,1] => [1,1] => 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => [3,1,1] => [1,1] => 1
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 3
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => [3,2,1] => [2,1] => 1
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => [3,3,1] => [3,1] => 2
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 2
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => [3,2,2] => [2,2] => 2
[1,3,1,3] => [[5,3,3,1],[2,2]] => [2,2] => [2] => 2
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => [3,3,2] => [3,2] => 4
[1,3,2,2] => [[5,4,3,1],[3,2]] => [3,2] => [2] => 2
[1,3,3,1] => [[5,5,3,1],[4,2]] => [4,2] => [2] => 2
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => [3,3,3] => [3,3] => 5
[1,4,1,2] => [[5,4,4,1],[3,3]] => [3,3] => [3] => 3
[1,4,2,1] => [[5,5,4,1],[4,3]] => [4,3] => [3] => 3
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [1,1,1,1,1,1] => [1,1,1,1,1] => 4
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => [2,1,1,1,1] => [1,1,1,1] => 3
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => [2,2,1,1,1] => [2,1,1,1] => 3
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 2
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => [3,1,1,1] => [1,1,1] => 2
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => [2,2,2,1,1] => [2,2,1,1] => 4
[2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 2
[2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]] => [3,2,1,1] => [2,1,1] => 2
[2,1,2,3] => [[5,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 1
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => [3,3,1,1] => [3,1,1] => 2
[2,1,3,2] => [[5,4,2,2],[3,1,1]] => [3,1,1] => [1,1] => 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => [2,2,2,2,1] => [2,2,2,1] => 5
[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 3
[2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]] => [3,2,2,1] => [2,2,1] => 3
[2,2,1,3] => [[5,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 1
[2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]] => [3,3,2,1] => [3,2,1] => 2
[2,2,2,2] => [[5,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 1
[2,2,3,1] => [[5,5,3,2],[4,2,1]] => [4,2,1] => [2,1] => 1
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => [3,3,3,1] => [3,3,1] => 7
[2,3,1,2] => [[5,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 2
[2,3,2,1] => [[5,5,4,2],[4,3,1]] => [4,3,1] => [3,1] => 2
[2,4,1,1] => [[5,5,5,2],[4,4,1]] => [4,4,1] => [4,1] => 3
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [2,2,2,2,2] => [2,2,2,2] => 8
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => [3,2,2,2] => [2,2,2] => 5
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 2
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => [3,3,2,2] => [3,2,2] => 7
[3,1,2,2] => [[5,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 2
[3,1,3,1] => [[5,5,3,3],[4,2,2]] => [4,2,2] => [2,2] => 2
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => [3,3,3,2] => [3,3,2] => 5
[3,2,1,2] => [[5,4,4,3],[3,3,2]] => [3,3,2] => [3,2] => 4
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search for individual values
searching the database for the individual values of this statistic
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
first row removal
Description
Removes the first entry of an integer partition
Map
inner shape
Description
The inner shape of a skew partition.
searching the database
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