Your data matches 153 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000911
(load all 3 compositions to match this statistic)
Mapping
St000911: Posets ⟶ 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
([(0,1),(0,2)],3)
 => 1
([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => 1
([],4)
 => 1
([(2,3)],4)
 => 2
([(1,2),(1,3)],4)
 => 1
([(0,1),(0,2),(0,3)],4)
 => 1
([(0,2),(0,3),(3,1)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
([(1,2),(2,3)],4)
 => 3
([(0,3),(3,1),(3,2)],4)
 => 1
([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(3,2)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => 2
([],5)
 => 1
([(3,4)],5)
 => 2
([(2,3),(2,4)],5)
 => 1
([(1,2),(1,3),(1,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(4,2)],5)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
([(2,3),(3,4)],5)
 => 3
([(1,4),(4,2),(4,3)],5)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => 1
([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(4,3)],5)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,3)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
Description
The number of maximal antichains of maximal size in a poset.
Matching statistic: St001616
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001616: 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
([(0,1),(0,2)],3)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 1
Description
The number of neutral elements in a lattice. An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St000363
(load all 2 compositions to match this statistic)
Mapping
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([(0,1)],2)
 => ([],2)
 => 1
([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => 1
([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 2
([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => 1
([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,2),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => 1
([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(3,4)],5)
 => 2
([(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(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),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
Description
The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St000228
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([],2)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([],3)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [3,1]
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => [2]
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => [1]
 => 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000550
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(2,3)],6)
 => ([(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)
 => ? ∊ {8,8,8,9}
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {8,8,8,9}
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
Mp00206: Posets —antichains of maximal size⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => 2
([],3)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
([],5)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(2,3)],6)
 => ([(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)
 => ? ∊ {8,8,8,9}
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {8,8,8,9}
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: St000479
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(2,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
Description
The Ramsey number of a graph. This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
Matching statistic: St001318
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001318: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(2,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Matching statistic: St001321
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001321: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(2,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
Description
The number of vertices of the largest induced subforest of a graph.
Matching statistic: St001342
Mapping
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001342: Graphs ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
([],1)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,2),(1,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? ∊ {8,8,8,9}
([(0,5),(1,4),(2,3)],6)
 => ([(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)
 => ([(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)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? ∊ {8,8,8,9}
([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {8,8,8,9}
Description
The number of vertices in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
The following 143 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001622The number of join-irreducible elements of a lattice. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001717The largest size of an interval in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000993The multiplicity of the largest part of an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000070The number of antichains in a poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001118The acyclic chromatic index of a graph. St000656The number of cuts of a poset. St001571The Cartan determinant of the integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000456The monochromatic index of 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. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000678The number of up steps after the last double rise of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001280The number of parts of an integer partition that are at least two. St001890The maximum magnitude of the Möbius function of a poset. St001924The number of cells in an integer partition whose arm and leg length coincide. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St001432The order dimension of the partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St001360The number of covering relations in Young's lattice below a partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001060The distinguishing index of a graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001877Number of indecomposable injective modules with projective dimension 2. 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 $eA$. 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 $eA$. St001481The minimal height of a peak of a Dyck path. St000120The number of left tunnels of a Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. 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. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. 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 $eA$. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000344The number of strongly connected outdegree sequences of a graph. St000379The number of Hamiltonian cycles in a graph. St001368The number of vertices of maximal degree in a graph.