Identifier
-
Mp00214:
Semistandard tableaux
—subcrystal⟶
Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[[1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1],[2]] => ([],1) => [1] => [1,0] => 0
[[1],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2]] => ([],1) => [1] => [1,0] => 0
[[1,2],[2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1],[2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2]] => ([],1) => [1] => [1,0] => 0
[[1,1,2],[2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,2]] => ([],1) => [1] => [1,0] => 0
[[1],[2],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,2],[2],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,2],[2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2]] => ([],1) => [1] => [1,0] => 0
[[1,1,2],[2,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1],[2],[3],[4]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,2],[2],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1],[2,3],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,2],[2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,2],[2,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2,2]] => ([],1) => [1] => [1,0] => 0
[[1],[2],[3],[5]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,2],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2],[3],[4]] => ([],1) => [1] => [1,0] => 0
[[1,2],[2],[3],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,2],[2],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1],[2,3],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,2],[2,2],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,2],[3,3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,1,1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,1,2],[2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,2],[2,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2,2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,2],[2,2,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2],[3],[5]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1],[2],[3],[4],[5]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1],[2],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2],[3],[4]] => ([],1) => [1] => [1,0] => 0
[[1,1,2],[2],[3],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,2],[3,4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1],[2,2],[3],[4]] => ([],1) => [1] => [1,0] => 0
[[1,1],[2,3],[3],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,2],[2],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1],[2,3],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,2],[2,2],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2,2],[3]] => ([],1) => [1] => [1,0] => 0
[[1,1,1],[2,2,3],[3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1],[2,2],[3,3]] => ([],1) => [1] => [1,0] => 0
[[1,1,2],[2,2],[3,3]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1,1,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1,1],[2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,1,1,2],[2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[2,2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,1,2],[2,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,2,2]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,2],[2,2,2]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2,2,2]] => ([],1) => [1] => [1,0] => 0
[[1]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1],[2,2,2],[3,3],[4]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,2],[2,2,2],[3,3],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2,3],[3,3],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1],[2,2,2],[3,4],[4]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]] => ([],1) => [1] => [1,0] => 0
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]] => ([(0,1)],2) => [2] => [1,0,1,0] => 1
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
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
subcrystal
Description
The underlying poset of the subcrystal obtained by applying the raising operators to a semistandard tableau.
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