- FindStat

Identifier

Mp00193: Lattices —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000480: Integer partitions ⟶ ℤ

Values

([],1) => ([],1) => [2] => [1,1] => 0
([(0,1)],2) => ([(0,1)],2) => [3] => [1,1,1] => 0
([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => [4] => [1,1,1,1] => 0
([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => [2,2,1,1] => 1
([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => [5] => [1,1,1,1,1] => 0
([(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] => 1
([(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] => 0
([(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] => 1
([(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] => 0
([(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] => 1
([(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] => 1
([(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] => 1
([(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] => 0
([(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] => 1
([(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] => 1
([(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] => 1
([(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] => 1
([(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] => 1
([(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] => 1
([(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] => 1
([(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] => 0
([(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] => 0
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => [6,2,2,2] => [4,4,1,1,1,1] => 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] => 1
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => [6,3,3] => [3,3,3,1,1,1] => 1
([(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
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => [6,2,2,2] => [4,4,1,1,1,1] => 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] => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => [6,2,2,2] => [4,4,1,1,1,1] => 1
([(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] => 1
([(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] => 0
([(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] => 1
([(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] => 0
([(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] => 0
([(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] => 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] => 0
([(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] => 0
([(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] => 0
([(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] => 0
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => [6,3,3] => [3,3,3,1,1,1] => 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] => 1
([(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] => 1
([(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] => 1

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$F_{1} = 1$

$F_{2} = 1$

$F_{3} = 1$

$F_{4} = 1 + q$

$F_{5} = 2 + 3\ q$

Description

The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for

$n\neq 2$ ) can cover is

$\frac{1}{2}(\sqrt{1+8n}-3)$
and an element which covers this number of elements is given by

$(c+i,c,c-1,\dots,3,2,1)$ , where

$1\leq i\leq c+2$ .

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.

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.