Identifier
-
Mp00313:
Integer partitions
—Glaisher-Franklin inverse⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤ
Values
[1] => [1] => [1,0] => 10 => 1
[2] => [1,1] => [1,1,0,0] => 1100 => 1
[1,1] => [2] => [1,0,1,0] => 1010 => 0
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Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let An=K[x]/(xn).
We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
Let An=K[x]/(xn).
We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
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
to binary word
Description
Return the Dyck word as binary word.
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