- FindStat

Your data matches 258 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000972
(load all 3 compositions to match this statistic)

Mapping

St000972: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The composition number of a graph. This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.

Matching statistic: St001474
(load all 3 compositions to match this statistic)

Mapping

St001474: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).

Matching statistic: St000550

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of modular elements of a lattice. A pair

$(x, y)$ of elements of a lattice

$L$ is a modular pair if for every

$z\geq y$ we have that

$(y\vee x) \wedge z = y \vee (x \wedge z)$ . An element

$x$ is left-modular if

$(x, y)$ is a modular pair for every

$y\in L$ , and is modular if both

$(x, y)$ and

$(y, x)$ are modular pairs for every

$y\in L$ .

Matching statistic: St000551

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left modular elements of a lattice. A pair

$(x, y)$ of elements of a lattice

$L$ is a modular pair if for every

$z\geq y$ we have that

$(y\vee x) \wedge z = y \vee (x \wedge z)$ . An element

$x$ is left-modular if

$(x, y)$ is a modular pair for every

$y\in L$ .

Matching statistic: St001706
(load all 2 compositions to match this statistic)

Mapping

Mp00156: Graphs —line graph⟶ Graphs
St001706: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of closed sets in a graph. A subset

$S$ of the set of vertices is a closed set, if for any pair of distinct elements of

$S$ the intersection of the corresponding neighbourhoods is a subset of

$S$ :

$\forall a, b\in S: N(a)\cap N(b) \subseteq S.$

Matching statistic: St001619

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001619: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-isomorphic sublattices of a lattice.

Matching statistic: St001666

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001666: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-isomorphic subposets of a lattice which are lattices.

Matching statistic: St000189

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000189: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of elements in the poset.

Matching statistic: St001717

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest size of an interval in a poset.

Matching statistic: St001300

Mapping

Mp00266: Graphs —connected vertex partitions⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.

The following 248 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000418The number of Dyck paths that are weakly below a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St001592The maximal number of simple paths between any two different vertices of a graph. St000656The number of cuts of a poset. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001279The sum of the parts of an integer partition that are at least two. St000395The sum of the heights of the peaks of a Dyck path. St000981The length of the longest zigzag subpath. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St000014The number of parking functions supported by a Dyck path. St000026The position of the first return of a Dyck path. St000543The size of the conjugacy class of a binary word. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001808The box weight or horizontal decoration of a Dyck path. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001930The weak major index of a binary word. St001956The comajor index for set-valued two-row standard Young tableaux. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000012The area of a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001161The major index north count of a Dyck path. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000379The number of Hamiltonian cycles in a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000806The semiperimeter of the associated bargraph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000477The weight of a partition according to Alladi. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000744The length of the path to the largest entry in a standard Young tableau. St000939The number of characters of the symmetric group whose value on the partition is positive. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001128The exponens consonantiae of a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St000509The diagonal index (content) of a partition. St000438The position of the last up step in a Dyck path. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001501The dominant dimension of magnitude 1 Nakayama algebras. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000467The hyper-Wiener index of a connected graph. St001281The normalized isoperimetric number of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001877Number of indecomposable injective modules with projective dimension 2. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000474Dyson's crank of a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001651The Frankl number of a lattice. St000699The toughness times the least common multiple of 1,. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001845The number of join irreducibles minus the rank of a lattice. St001626The number of maximal proper sublattices of a lattice.