- FindStat

Identifier

Mp00243: Graphs —weak duplicate order⟶ Posets
St001902: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

([(0,3),(1,2)],4) => ([],4) => 12

([(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) => 6

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

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

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

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 12

([],5) => ([],1) => 0

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

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

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

([(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) => 12

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

([(0,1),(2,4),(3,4)],5) => ([],4) => 12

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

([(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) => 5

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

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

([(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) => 6

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

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

([(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) => 6

([(0,4),(1,3),(2,3),(2,4)],5) => ([(2,3),(2,4)],5) => 10

([(0,1),(2,3),(2,4),(3,4)],5) => ([],5) => 20

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(3,4)],5) => 12

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([],5) => 20

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([],5) => 20

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

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

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

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 12

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 10

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 10

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

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],4) => 12

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],5) => 20

([],6) => ([],1) => 0

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

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

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

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

([(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) => 12

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

([(1,2),(3,5),(4,5)],6) => ([(0,4),(1,4),(2,4),(3,4)],5) => 12

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

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

([(0,1),(2,5),(3,5),(4,5)],6) => ([],4) => 12

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

([(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) => 5

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

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

([(0,5),(1,5),(2,4),(3,4)],6) => ([],4) => 12

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

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(2,3),(2,4)],5) => 10

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

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

([(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) => 5

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

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

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(2,3),(2,4)],5) => 10

([(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) => 6

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

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

([(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) => 6

([(0,5),(1,4),(2,3)],6) => ([],6) => 30

([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 10

([(0,1),(2,5),(3,4),(4,5)],6) => ([(2,5),(3,4)],6) => 12

([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 20

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

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 12

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 17

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4)],6) => 10

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 20

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 17

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 20

([(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) => 6

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 17

>>> Load all 1134 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of potential covers of a poset.
A potential cover is a pair of uncomparable elements $(x, y)$ which can be added to the poset without adding any other relations.
For example, let $P$ be the disjoint union of a single relation $(1, 2)$ with the one element poset $0$. Then the relation $(0, 1)$ cannot be added without adding also $(0, 2)$, however, the relations $(0, 2)$ and $(1, 0)$ are potential covers.

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.