Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤ
Values
[1,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[2,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[1,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[3,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[2,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[2,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[4,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[3,3] => [3] => [3] => [1,0,1,0,1,0] => 3
[3,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[3,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[2,2,2] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 3
[2,2,1,1] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[1,1,1,1,1,1] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[5,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[4,3] => [3] => [3] => [1,0,1,0,1,0] => 3
[4,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[4,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[3,3,1] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[3,2,2] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 3
[3,2,1,1] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[2,2,2,1] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[2,2,1,1,1] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[2,1,1,1,1,1] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[6,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[5,3] => [3] => [3] => [1,0,1,0,1,0] => 3
[5,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[5,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[4,4] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[4,3,1] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[4,2,2] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 3
[4,2,1,1] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[3,3,2] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3
[3,3,1,1] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[3,2,2,1] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[3,2,1,1,1] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[3,1,1,1,1,1] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[2,2,2,1,1] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3
[2,2,1,1,1,1] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[7,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[6,3] => [3] => [3] => [1,0,1,0,1,0] => 3
[6,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[6,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[5,4] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[5,3,1] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[5,2,2] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 3
[5,2,1,1] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[4,4,1] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 4
[4,3,2] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3
[4,3,1,1] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[4,2,2,1] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[4,2,1,1,1] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[4,1,1,1,1,1] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[3,3,1,1,1] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 3
[3,2,2,1,1] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3
[3,2,1,1,1,1] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[8,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[7,3] => [3] => [3] => [1,0,1,0,1,0] => 3
[7,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[7,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[6,4] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[6,3,1] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[6,2,2] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 3
[6,2,1,1] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[5,5] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
[5,4,1] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 4
[5,3,2] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3
[5,3,1,1] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[5,2,2,1] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[5,2,1,1,1] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[5,1,1,1,1,1] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[4,3,1,1,1] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 3
[4,2,2,1,1] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3
[4,2,1,1,1,1] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[3,3,2,2] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[3,3,1,1,1,1] => [3,1,1,1,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[2,2,2,2,2] => [2,2,2,2] => [4,4] => [1,1,1,0,1,0,1,0,0,0] => 3
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => [2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 2
[9,1,1] => [1,1] => [2] => [1,0,1,0] => 2
[8,3] => [3] => [3] => [1,0,1,0,1,0] => 3
[8,2,1] => [2,1] => [1,1,1] => [1,1,0,1,0,0] => 2
[8,1,1,1] => [1,1,1] => [2,1] => [1,0,1,1,0,0] => 2
[7,4] => [4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 3
[7,3,1] => [3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 3
[7,2,2] => [2,2] => [4] => [1,0,1,0,1,0,1,0] => 3
[7,2,1,1] => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3
[6,5] => [5] => [5] => [1,0,1,0,1,0,1,0,1,0] => 3
[6,4,1] => [4,1] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 4
[6,3,2] => [3,2] => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3
[6,3,1,1] => [3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 2
[6,2,2,1] => [2,2,1] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[6,2,1,1,1] => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4
[6,1,1,1,1,1] => [1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 2
[5,3,1,1,1] => [3,1,1,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 3
[5,2,2,1,1] => [2,2,1,1] => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 3
[5,2,1,1,1,1] => [2,1,1,1,1] => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[4,3,2,2] => [3,2,2] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 3
[4,3,1,1,1,1] => [3,1,1,1,1] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 2
>>> Load all 257 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
Glaisher-Franklin inverse
Description
The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
first row removal
Description
Removes the first entry of an integer partition
searching the database
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