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St000189: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> 2 = 3 - 1
([(0,1)],2)
=> 2 = 3 - 1
([(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
Description
The number of elements in the poset.
Mp00195: Posets order idealsLattices
St001875: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 6
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of simple modules with projective dimension at most 1.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> 2 = 3 - 1
([(0,1)],2)
=> [2]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> [3]
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 7 - 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00198: Posets incomparability graphGraphs
St001318: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 7 - 1
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
Mp00198: Posets incomparability graphGraphs
St001321: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 7 - 1
Description
The number of vertices of the largest induced subforest of a graph.
Mp00195: Posets order idealsLattices
St001615: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
Description
The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Mp00195: Posets order idealsLattices
St001617: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
Description
The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying $$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Mp00195: Posets order idealsLattices
St001622: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Mp00198: Posets incomparability graphGraphs
St001672: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1)],2)
=> 2 = 3 - 1
([(0,1)],2)
=> ([],2)
=> 2 = 3 - 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 3 = 4 - 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 4 - 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 4 = 5 - 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 5 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 4 = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 6 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5 = 6 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 6 = 7 - 1
Description
The restrained domination number of a graph. This is the minimal size of a set of vertices $D$ such that every vertex not in $D$ is adjacent to a vertex in $D$ and to a vertex not in $D$.
Mp00198: Posets incomparability graphGraphs
Mp00259: Graphs vertex additionGraphs
St000479: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],2)
=> ([(0,1)],2)
=> ([(1,2)],3)
=> 3
([(0,1)],2)
=> ([],2)
=> ([],3)
=> 3
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 4
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4
([(0,2),(2,1)],3)
=> ([],3)
=> ([],4)
=> 4
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(2,3)],4)
=> 4
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 5
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(3,4)],5)
=> 5
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(3,4)],5)
=> 5
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(3,4)],5)
=> 5
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 5
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],5)
=> 5
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(4,5)],6)
=> 6
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(4,5)],6)
=> 6
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(4,5)],6)
=> 6
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],6)
=> 6
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(4,5)],6)
=> 6
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([],7)
=> 7
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]
The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000718The largest Laplacian eigenvalue of a graph if it is integral. St000806The semiperimeter of the associated bargraph. St001342The number of vertices in the center of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St000171The degree of the graph. St000293The number of inversions of a binary word. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000636The hull number of a graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001034The area of the parallelogram polyomino associated with the Dyck path. St001120The length of a longest path in a graph. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000273The domination number of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000532The total number of rook placements on a Ferrers board. St000543The size of the conjugacy class of a binary word. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001437The flex of a binary word. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001829The common independence number of a graph. St000018The number of inversions of a permutation. St000144The pyramid weight of the Dyck path. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000246The number of non-inversions of a permutation. St000259The diameter of a connected graph. St000290The major index of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000507The number of ascents of a standard tableau. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000734The last entry in the first row of a standard tableau. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000922The minimal number such that all substrings of this length are unique. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001360The number of covering relations in Young's lattice below a partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001959The product of the heights of the peaks of a Dyck path. St000681The Grundy value of Chomp on Ferrers diagrams. St000744The length of the path to the largest entry in a standard Young tableau. St000820The number of compositions obtained by rotating the composition. St000921The number of internal inversions of a binary word. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001759The Rajchgot index of a permutation. St000741The Colin de Verdière graph invariant. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000719The number of alignments in a perfect matching. St001645The pebbling number of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001820The size of the image of the pop stack sorting operator. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St001720The minimal length of a chain of small intervals in a lattice. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000080The rank of the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001118The acyclic chromatic index of a graph. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000455The second largest eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000454The largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. 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. St000327The number of cover relations in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000264The girth of a graph, which is not a tree. St000422The energy of a graph, if it is integral. 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. St001877Number of indecomposable injective modules with projective dimension 2. St000699The toughness times the least common multiple of 1,.