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Mp00206: Posets antichains of maximal sizeLattices
St000550: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(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
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(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
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> 2
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 $y\in L$.
Mp00206: Posets antichains of maximal sizeLattices
St000551: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(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
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(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
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> 2
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 $y\in L$.
Mp00206: Posets antichains of maximal sizeLattices
St001616: Lattices ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
([(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
([(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(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
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> 2
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 $x, y \in L$.
Mp00206: Posets antichains of maximal sizeLattices
Mp00263: Lattices join irreduciblesPosets
St000070: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2)],3)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(1,2)],3)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(1,2)],3)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],2)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,1)],2)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([],1)
=> 2
Description
The number of antichains in a poset. An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable. An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
St000656: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(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)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(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)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
Description
The number of cuts of a poset. A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
St001717: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(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)
=> ([(0,1)],2)
=> 2
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(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)
=> 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
Description
The largest size of an interval in a poset.
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
St001300: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 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 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 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
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 5 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
([(0,3),(1,2),(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
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000228
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(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
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(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
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 2
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000479
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
Mp00198: Posets incomparability graphGraphs
St000479: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(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
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(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
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
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
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
Mp00198: Posets incomparability graphGraphs
St001318: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(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
([(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(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
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,3),(1,4),(4,2)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
([(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(1,3),(1,4),(1,5),(5,2)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
Description
The number of vertices of the largest induced subforest with the same number of connected components of a graph.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. 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. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000363The number of minimal vertex covers of a graph. St000422The energy of a graph, if it is integral. St000911The number of maximal antichains of maximal size in a poset. St000993The multiplicity of the largest part of an integer 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. 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.