- FindStat

Your data matches 114 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000914
(load all 3 compositions to match this statistic)

Mapping

St000914: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The sum of the values of the Möbius function of a poset. The Möbius function $\mu$ of a finite poset is defined as $$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$ Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is $$ \sum_{x\leq y} \mu(x,y). $$ If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components. This statistic is also called the magnitude of a poset.

Matching statistic: St001363
(load all 5 compositions to match this statistic)

Mapping

Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Euler characteristic of a graph according to Knill. This is $$\sum_{k\geq 1} (-1)^{k-1} c_k,$$ where $c_k$ is the number of cliques with $k$ vertices.

Matching statistic: St000068

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000068: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 60%●distinct values known / distinct values provided: 30%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The number of minimal elements in a poset.

Matching statistic: St000181

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000181: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 60%●distinct values known / distinct values provided: 30%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The number of connected components of the Hasse diagram for the poset.

Matching statistic: St000298

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000298: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 60%●distinct values known / distinct values provided: 20%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

Matching statistic: St000307

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000307: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 60%●distinct values known / distinct values provided: 40%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.

Matching statistic: St000527

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000527: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 60%●distinct values known / distinct values provided: 30%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St000908

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000908: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 60%●distinct values known / distinct values provided: 30%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 2
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The length of the shortest maximal antichain in a poset.

Matching statistic: St000909

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St000909: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 60%●distinct values known / distinct values provided: 40%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The number of maximal chains of maximal size in a poset.

Matching statistic: St001268

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St001268: Posets ⟶ ℤResult quality: 40% ●values known / values provided: 60%●distinct values known / distinct values provided: 40%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? = 2
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,2,3}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {0,1,1,2,2,2,3,4}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(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)
 => 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,4),(1,2),(1,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(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)
 => 3
([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 1
([(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)
 => 1
([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => 2
([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,4,5}
([],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(4,5)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(3,4),(3,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(2,3),(2,4),(2,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(1,2),(1,3),(1,4),(1,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(1,3),(1,4),(1,5),(5,2)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
 => ([],1)
 => ([],0)
 => ? ∊ {-3,-2,-2,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,5,6}
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
 => ([(0,1)],2)
 => ([],1)
 => 1

Description

The size of the largest ordinal summand in the poset. The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$. This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.

The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001890The maximum magnitude of the Möbius function of a poset. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000100The number of linear extensions of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000668The least common multiple of the parts of the partition. 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. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001118The acyclic chromatic index of a graph. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer 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. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. 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. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000120The number of left tunnels of a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St000455The second largest eigenvalue of a graph if it is integral.