Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000550
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
Description
The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Matching statistic: St001616
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
Description
The number of neutral elements in a lattice. An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St001720
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 1
Description
The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001615
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
Description
The number of join prime elements of a lattice. An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
Description
The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying $$ v(a\vee b) + v(a\wedge b) = v(a) + v(b). $$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001619
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001619: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
Description
The number of non-isomorphic sublattices of a lattice.
Matching statistic: St001622
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 0 = 1 - 1
Description
The number of join-irreducible elements of a lattice. An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St001666
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001666: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,0]
 => ([],2)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => ([],3)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([],1)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => 2 = 1 + 1
Description
The number of non-isomorphic subposets of a lattice which are lattices.
Matching statistic: St000382
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => [2] => 2
[1,1,0,0]
 => [2,1] => [2,1] => [1,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => [3] => 3
[1,0,1,1,0,0]
 => [1,3,2] => [3,1,2] => [1,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => [1,2] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => [2,1] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => [1,1,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => [4] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [4,3,1,2] => [1,1,2] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [4,2,3,1] => [1,2,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => [1,2,1] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => [2,1,1] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => [1,4] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [4,5,1,2,3] => [2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [3,5,1,2,4] => [2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,4,1,2,5] => [2,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [3,4,5,1,2] => [3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [5,3,4,1,2] => [1,2,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [4,3,5,1,2] => [1,2,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [4,5,3,1,2] => [2,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,4] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,5,1,3,4] => [2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,4,1,3,5] => [2,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,4,5,1,3] => [3,2] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [5,2,4,1,3] => [1,2,2] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => [2,3] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,5,1,4] => [3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => [3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [5,2,3,4,1] => [1,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [4,2,3,1,5] => [1,2,2] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,2,3,5,1] => [1,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [4,5,2,3,1] => [2,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [5,4,2,3,1] => [1,1,2,1] => 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,6,5,8,7,4,3,2] => [6,8,5,7,4,3,1,2] => ? => ? = 2
[1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,3,7,8,6,5,4,1] => [7,8,6,5,2,3,4,1] => ? => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
 => [3,2,4,5,6,1,8,7] => ? => ? => ? = 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0,1,0]
 => [4,5,3,7,6,2,1,8] => [4,5,7,3,6,2,1,8] => ? => ? = 3
Description
The first part of an integer composition.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000297The number of leading ones in a binary word. St000011The number of touch points (or returns) of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000069The number of maximal elements of a poset. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000363The number of minimal vertex covers of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000909The number of maximal chains of maximal size in a poset. St000617The number of global maxima of a Dyck path. St000911The number of maximal antichains of maximal size in a poset. St000883The number of longest increasing subsequences of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001651The Frankl number of a lattice. St000054The first entry of the permutation. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000061The number of nodes on the left branch of a binary tree. St001875The number of simple modules with projective dimension at most 1. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001267The length of the Lyndon factorization of the binary word. St001904The length of the initial strictly increasing segment of a parking function.