- FindStat

Your data matches 590 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001613
(load all 4 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.

Matching statistic: St001615
(load all 5 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of join prime elements of a lattice. An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

Matching statistic: St001617
(load all 5 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice

$L$ is a function

$v:L\mapsto\mathbb R$ satisfying

$v(a\vee b) + v(a\wedge b) = v(a) + v(b).$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

Matching statistic: St001621
(load all 4 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

Matching statistic: St001622
(load all 3 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .

Matching statistic: St001623
(load all 3 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001623: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of doubly irreducible elements of a lattice. An element

$d$ of a lattice

$L$ is '''doubly irreducible''' if it is both join and meet irreducible. That means,

$d$ is neither the least nor the greatest element of

$L$ and if

$d=x\vee y$ or

$d=x\wedge y$ , then

$d\in\{x,y\}$ for all

$x,y\in L$ . In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.

Matching statistic: St001626
(load all 3 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of maximal proper sublattices of a lattice.

Matching statistic: St001677
(load all 7 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001677: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of non-degenerate subsets of a lattice whose meet is the bottom element. A subset whose meet is the bottom element is non-degenerate, if it neither contains the bottom, nor the top element of the lattice.

Matching statistic: St001845
(load all 15 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of join irreducibles minus the rank of a lattice. A lattice is join-extremal, if this statistic is

$0$ .

Matching statistic: St001846
(load all 4 compositions to match this statistic)

Mapping

Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(5,4),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2)],8)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,6),(5,1),(5,2),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,2),(4,6),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,6),(3,7),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(3,7),(4,6),(5,1),(6,7)],8)
 => ([],1)
 => 0
([(0,2),(0,3),(0,5),(1,7),(2,7),(3,6),(4,1),(4,6),(5,4),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(1,7),(2,7),(3,6),(4,2),(5,1),(5,6),(6,7)],8)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,1),(7,2),(7,3),(7,4)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,3),(7,1),(7,2),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(4,8),(5,8),(6,1),(6,2),(6,3),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,2),(6,3),(7,1),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,7),(1,8),(2,8),(3,8),(4,8),(5,2),(6,1),(7,5),(7,6)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,7),(3,8),(4,8),(5,1),(5,7),(6,2),(6,5),(7,8)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,5),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2)],9)
 => ([],1)
 => 0
([(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,1),(7,2),(7,3)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,7),(8,1)],9)
 => ([],1)
 => 0
([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,7),(8,1),(8,2)],9)
 => ([],1)
 => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(8,7)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,7),(6,1),(7,8)],9)
 => ([],1)
 => 0
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(6,8),(8,7)],9)
 => ([],1)
 => 0

Description

The number of elements which do not have a complement in the lattice. A complement of an element

$x$ in a lattice is an element

$y$ such that the meet of

$x$ and

$y$ is the bottom element and their join is the top element.

The following 580 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001616The number of neutral elements in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001820The size of the image of the pop stack sorting operator. St001833The number of linear intervals in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001619The number of non-isomorphic sublattices of a lattice. St001620The number of sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000069The number of maximal elements of a poset. St000080The rank of the poset. St000632The jump number of the poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St001902The number of potential covers of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000189The number of elements in the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000527The width of the poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001268The size of the largest ordinal summand in the poset. St001343The dimension of the reduced incidence algebra of a poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St001779The order of promotion on the set of linear extensions of a poset. St001813The product of the sizes of the principal order filters in a poset. St001815The number of order preserving surjections from a poset to a total order. St000070The number of antichains in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000180The number of chains of a poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001909The number of interval-closed sets of a poset. St001709The number of homomorphisms to the three element chain of a poset. St000081The number of edges of a graph. St000095The number of triangles of a graph. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000145The Dyson rank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000274The number of perfect matchings of a graph. St000302The determinant of the distance matrix of a connected graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000311The number of vertices of odd degree in a graph. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000350The sum of the vertex degrees of a graph. St000351The determinant of the adjacency matrix of a graph. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000377The dinv defect of an integer partition. St000387The matching number of a graph. St000403The Szeged index minus the Wiener index of a graph. St000422The energy of a graph, if it is integral. St000447The number of pairs of vertices of a graph with distance 3. St000448The number of pairs of vertices of a graph with distance 2. St000449The number of pairs of vertices of a graph with distance 4. St000454The largest eigenvalue of a graph if it is integral. St000465The first Zagreb index of a graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000571The F-index (or forgotten topological index) of a graph. St000637The length of the longest cycle in a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000778The metric dimension of a graph. St000915The Ore degree of a graph. St000944The 3-degree of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000995The largest even part of an integer partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001214The aft of an integer partition. St001248Sum of the even parts of a partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001349The number of different graphs obtained from the given graph by removing an edge. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001362The normalized Knill dimension of a graph. St001374The Padmakar-Ivan index of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001391The disjunction number of a graph. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001541The Gini index of an integer partition. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001649The length of a longest trail in a graph. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001692The number of vertices with higher degree than the average degree in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001703The villainy of a graph. St001708The number of pairs of vertices of different degree in a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001736The total number of cycles in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001743The discrepancy of a graph. St001764The number of non-convex subsets of vertices in a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001783The number of odd automorphisms of a graph. St001792The arboricity of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001871The number of triconnected components of a graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000003The number of standard Young tableaux of the partition. St000010The length of the partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000086The number of subgraphs. St000087The number of induced subgraphs. St000088The row sums of the character table of the symmetric group. St000093The cardinality of a maximal independent set of vertices of a graph. St000096The number of spanning trees of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000137The Grundy value of an integer partition. St000146The Andrews-Garvan crank of a partition. St000147The largest part of an integer partition. St000148The number of odd parts of a partition. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000172The Grundy number of a graph. St000179The product of the hook lengths of the integer partition. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000228The size of a partition. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000268The number of strongly connected orientations of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000271The chromatic index of a graph. St000273The domination number of a graph. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000321The number of integer partitions of n that are dominated by an integer partition. St000343The number of spanning subgraphs of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000349The number of different adjacency matrices of a graph. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000474Dyson's crank of a partition. St000475The number of parts equal to 1 in a partition. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000517The Kreweras number of an integer partition. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000553The number of blocks of a graph. St000618The number of self-evacuating tableaux of given shape. St000636The hull number of a graph. St000644The number of graphs with given frequency partition. St000667The greatest common divisor of the parts of the partition. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000722The number of different neighbourhoods in a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St000948The chromatic discriminant of a graph. St000972The composition number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001029The size of the core of a graph. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001057The Grundy value of the game of creating an independent set in a graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001072The evaluation of the Tutte polynomial of the graph at x and y equal to 3. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001116The game chromatic number of a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001127The sum of the squares of the parts of a partition. St001129The product of the squares of the parts of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001261The Castelnuovo-Mumford regularity of a graph. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. 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. St001286The annihilation number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001303The number of dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001366The maximal multiplicity of a degree of a vertex of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001368The number of vertices of maximal degree in a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001386The number of prime labellings of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001441The number of non-empty connected induced subgraphs of a graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001463The number of distinct columns in the nullspace of a graph. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001546The number of monomials in the Tutte polynomial of a graph. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001691The number of kings in a graph. St001694The number of maximal dissociation sets in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001716The 1-improper chromatic number of a graph. St001725The harmonious chromatic number of a graph. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001763The Hurwitz number of an integer partition. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. 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. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001802The number of endomorphisms of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St001963The tree-depth of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000759The smallest missing part in an integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001706The number of closed sets in a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001762The number of convex subsets of vertices in a graph. St001814The number of partitions interlacing the given partition. St001834The number of non-isomorphic minors of a graph. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St001095The number of non-isomorphic posets with precisely one further covering relation. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001281The normalized isoperimetric number of a graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000264The girth of a graph, which is not a tree. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000717The number of ordinal summands of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001668The number of points of the poset minus the width of the poset. St000680The Grundy value for Hackendot on posets. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St001060The distinguishing index of a graph. St001637The number of (upper) dissectors of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph.