- FindStat

Identifier

Mp00243: Graphs —weak duplicate order⟶ Posets
St000914: Posets ⟶ ℤ

Values

([(0,1)],2) => ([],2) => 2

([(1,2)],3) => ([(0,2),(1,2)],3) => 1

([(0,2),(1,2)],3) => ([],2) => 2

([(0,1),(0,2),(1,2)],3) => ([],3) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.

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.