Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤ
Values
[1,1,1] => [[1,1,1],[]] => [1,1,1] => [1,1] => 2
[2,1] => [[2,2],[1]] => [2,2] => [2] => 1
[1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1] => [1,1,1] => 3
[1,1,2] => [[2,1,1],[]] => [2,1,1] => [1,1] => 2
[1,2,1] => [[2,2,1],[1]] => [2,2,1] => [2,1] => 1
[2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => [2,2] => 2
[2,2] => [[3,2],[1]] => [3,2] => [2] => 1
[3,1] => [[3,3],[2]] => [3,3] => [3] => 1
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [1,1,1,1] => 4
[1,1,1,2] => [[2,1,1,1],[]] => [2,1,1,1] => [1,1,1] => 3
[1,1,2,1] => [[2,2,1,1],[1]] => [2,2,1,1] => [2,1,1] => 1
[1,1,3] => [[3,1,1],[]] => [3,1,1] => [1,1] => 2
[1,2,1,1] => [[2,2,2,1],[1,1]] => [2,2,2,1] => [2,2,1] => 2
[1,2,2] => [[3,2,1],[1]] => [3,2,1] => [2,1] => 1
[1,3,1] => [[3,3,1],[2]] => [3,3,1] => [3,1] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => [2,2,2] => 3
[2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => [2,2] => 2
[2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => [3,2] => 1
[2,3] => [[4,2],[1]] => [4,2] => [2] => 1
[3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => [3,3] => 2
[3,2] => [[4,3],[2]] => [4,3] => [3] => 1
[4,1] => [[4,4],[3]] => [4,4] => [4] => 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [1,1,1,1,1] => 5
[1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => [1,1,1,1] => 4
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => [2,1,1,1] => 1
[1,1,1,3] => [[3,1,1,1],[]] => [3,1,1,1] => [1,1,1] => 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [2,2,2,1,1] => [2,2,1,1] => 2
[1,1,2,2] => [[3,2,1,1],[1]] => [3,2,1,1] => [2,1,1] => 1
[1,1,3,1] => [[3,3,1,1],[2]] => [3,3,1,1] => [3,1,1] => 1
[1,1,4] => [[4,1,1],[]] => [4,1,1] => [1,1] => 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => [2,2,2,1] => 3
[1,2,1,2] => [[3,2,2,1],[1,1]] => [3,2,2,1] => [2,2,1] => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => [3,3,2,1] => [3,2,1] => 1
[1,2,3] => [[4,2,1],[1]] => [4,2,1] => [2,1] => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => [3,3,1] => 2
[1,3,2] => [[4,3,1],[2]] => [4,3,1] => [3,1] => 1
[1,4,1] => [[4,4,1],[3]] => [4,4,1] => [4,1] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => [2,2,2,2] => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => [2,2,2] => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [3,3,2,2] => [3,2,2] => 1
[2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => [2,2] => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [3,3,3,2] => [3,3,2] => 2
[2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => [3,2] => 1
[2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => [4,2] => 1
[2,4] => [[5,2],[1]] => [5,2] => [2] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [3,3,3,3] => [3,3,3] => 3
[3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => [3,3] => 2
[3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => [4,3] => 1
[3,3] => [[5,3],[2]] => [5,3] => [3] => 1
[4,1,1] => [[4,4,4],[3,3]] => [4,4,4] => [4,4] => 2
[4,2] => [[5,4],[3]] => [5,4] => [4] => 1
[5,1] => [[5,5],[4]] => [5,5] => [5] => 1
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => [1,1,1,1,1] => 5
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [2,2,1,1,1,1] => [2,1,1,1,1] => 1
[1,1,1,1,3] => [[3,1,1,1,1],[]] => [3,1,1,1,1] => [1,1,1,1] => 4
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [2,2,2,1,1,1] => [2,2,1,1,1] => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [3,2,1,1,1] => [2,1,1,1] => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [3,3,1,1,1] => [3,1,1,1] => 1
[1,1,1,4] => [[4,1,1,1],[]] => [4,1,1,1] => [1,1,1] => 3
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [2,2,2,2,1,1] => [2,2,2,1,1] => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [3,2,2,1,1] => [2,2,1,1] => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => [3,2,1,1] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [4,2,1,1] => [2,1,1] => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [3,3,3,1,1] => [3,3,1,1] => 2
[1,1,3,2] => [[4,3,1,1],[2]] => [4,3,1,1] => [3,1,1] => 1
[1,1,4,1] => [[4,4,1,1],[3]] => [4,4,1,1] => [4,1,1] => 1
[1,1,5] => [[5,1,1],[]] => [5,1,1] => [1,1] => 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [2,2,2,2,2,1] => [2,2,2,2,1] => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => [2,2,2,1] => 3
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [3,3,2,2,1] => [3,2,2,1] => 1
[1,2,1,3] => [[4,2,2,1],[1,1]] => [4,2,2,1] => [2,2,1] => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [3,3,3,2,1] => [3,3,2,1] => 2
[1,2,2,2] => [[4,3,2,1],[2,1]] => [4,3,2,1] => [3,2,1] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [4,4,2,1] => [4,2,1] => 1
[1,2,4] => [[5,2,1],[1]] => [5,2,1] => [2,1] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [3,3,3,3,1] => [3,3,3,1] => 3
[1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => [3,3,1] => 2
[1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => [4,3,1] => 1
[1,3,3] => [[5,3,1],[2]] => [5,3,1] => [3,1] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [4,4,4,1] => [4,4,1] => 2
[1,4,2] => [[5,4,1],[3]] => [5,4,1] => [4,1] => 1
[1,5,1] => [[5,5,1],[4]] => [5,5,1] => [5,1] => 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2] => [2,2,2,2,2] => 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [3,2,2,2,2] => [2,2,2,2] => 4
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [3,3,2,2,2] => [3,2,2,2] => 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => [2,2,2] => 3
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [3,3,3,2,2] => [3,3,2,2] => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => [3,2,2] => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [4,4,2,2] => [4,2,2] => 1
[2,1,4] => [[5,2,2],[1,1]] => [5,2,2] => [2,2] => 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [3,3,3,3,2] => [3,3,3,2] => 3
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [4,3,3,2] => [3,3,2] => 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [4,4,3,2] => [4,3,2] => 1
[2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => [3,2] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [4,4,4,2] => [4,4,2] => 2
[2,3,2] => [[5,4,2],[3,1]] => [5,4,2] => [4,2] => 1
[2,4,1] => [[5,5,2],[4,1]] => [5,5,2] => [5,2] => 1
[2,5] => [[6,2],[1]] => [6,2] => [2] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [3,3,3,3,3] => [3,3,3,3] => 4
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [4,3,3,3] => [3,3,3] => 3
>>> Load all 210 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The multiplicity of the largest part of an integer partition.
Map
outer shape
Description
The outer shape of the skew partition.
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
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!