- FindStat

Identifier

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001615: Lattices ⟶ ℤ

Values

([],1) => ([],1) => ([(0,1)],2) => 1
([],2) => ([],1) => ([(0,1)],2) => 1
([(0,1)],2) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([],3) => ([],1) => ([(0,1)],2) => 1
([(1,2)],3) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,2),(1,2)],3) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,1),(0,2),(1,2)],3) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([],4) => ([],1) => ([(0,1)],2) => 1
([(2,3)],4) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(1,3),(2,3)],4) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,3),(1,3),(2,3)],4) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,2),(0,3),(1,2),(1,3)],4) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([],5) => ([],1) => ([(0,1)],2) => 1
([(3,4)],5) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(1,4),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,4),(1,4),(2,4),(3,4)],5) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(2,3),(2,4),(3,4)],5) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([],6) => ([],1) => ([(0,1)],2) => 1
([(4,5)],6) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(2,5),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([],7) => ([],1) => ([(0,1)],2) => 1
([(5,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(3,6),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(2,6),(3,6),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(4,5),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(3,5),(3,6),(4,5),(4,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 3
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,3),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9) => 4
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3
([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([],3) => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8) => 3

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$F_{1} = q$

$F_{2} = q + q^{2}$

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

Description

The number of join prime elements of a lattice.
An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

Map

weak duplicate order

Description

The weak duplicate order of the de-duplicate of a graph.
Let

$G=(V, E)$ be a graph and let

$N=\{ N_v | v\in V\}$ be the set of (distinct) neighbourhoods of

$G$ .
This map yields the poset obtained by ordering

$N$ by reverse inclusion.

Map

order ideals

Description

The lattice of order ideals of a poset.
An order ideal

$\mathcal I$ in a poset

$P$ is a downward closed set, i.e.,

$a \in \mathcal I$ and

$b \leq a$ implies

$b \in \mathcal I$ . This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.