Identifier
-
Mp00193:
Lattices
—to poset⟶
Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤ
Values
([],1) => ([],1) => [2] => [1,1] => 1
([(0,1)],2) => ([(0,1)],2) => [3] => [1,1,1] => 1
([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => [4] => [1,1,1,1] => 1
([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => [2,2,1,1] => 2
([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => [5] => [1,1,1,1,1] => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [4,2,2,2] => [4,4,1,1] => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => [1,1,1,1,1,1,1,1] => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => [2,2,1,1,1] => 2
([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => [6] => [1,1,1,1,1,1] => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => [2,2,1,1,1] => 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => [8,2,2] => [3,3,1,1,1,1,1,1] => 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [5,2,2,2] => [4,4,1,1,1] => 2
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [6,2] => [2,2,1,1,1,1] => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => [2,2,2,2,1,1] => 2
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [6,2] => [2,2,1,1,1,1] => 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => [8,2,2] => [3,3,1,1,1,1,1,1] => 2
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [6,2] => [2,2,1,1,1,1] => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [5,2,2,2] => [4,4,1,1,1] => 2
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [5,3,3] => [3,3,3,1,1] => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => [2,2,2,2,2] => 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [7] => [1,1,1,1,1,1,1] => 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 1
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [7,2] => [2,2,1,1,1,1,1] => 2
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => [6,5] => [2,2,2,2,2,1] => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [6,2,2] => [3,3,1,1,1,1] => 2
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => [6,5,3,3] => [4,4,4,2,2,1] => 2
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [7,2] => [2,2,1,1,1,1,1] => 2
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [6,5] => [2,2,2,2,2,1] => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => [6,5,3,3] => [4,4,4,2,2,1] => 2
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => [7,4] => [2,2,2,2,1,1,1] => 2
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [7,2] => [2,2,1,1,1,1,1] => 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => [16] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => [14] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => [12] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [8] => [1,1,1,1,1,1,1,1] => 1
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => [7,4] => [2,2,2,2,1,1,1] => 2
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [7,2] => [2,2,1,1,1,1,1] => 2
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => [7,5] => [2,2,2,2,2,1,1] => 2
([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10) => [8,8] => [2,2,2,2,2,2,2,2] => 1
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Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
See Section 3.3.1 of [2].
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
to poset
Description
Return the poset corresponding to the lattice.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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