- FindStat

Identifier

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

Description

The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice 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.

Map

dual poset

Description

The dual of a poset.
The dual (or opposite) of a poset

$(\mathcal P,\leq)$ is the poset

$(\mathcal P^d,\leq_d)$ with

$x \leq_d y$ if

$y \leq x$ .