Your data matches 177 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 ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St000550: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
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 ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St000551: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
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
(load all 2 compositions to match this statistic)
Mapping
Mp00242: Dyck paths ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001616: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
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: St001626
Mapping
Mp00242: Dyck paths ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001626: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
Description
The number of maximal proper sublattices of a lattice.
Matching statistic: St001720
(load all 2 compositions to match this statistic)
Mapping
Mp00242: Dyck paths ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001720: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
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: St001875
Mapping
Mp00242: Dyck paths ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001875: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001615
Mapping
Mp00242: Dyck paths ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001615: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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 ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001617: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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 ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001619: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 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
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 7 = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 6 = 5 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 8 = 7 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 4 = 3 + 1
Description
The number of non-isomorphic sublattices of a lattice.
Matching statistic: St001622
Mapping
Mp00242: Dyck paths ā€”Hessenberg posetāŸ¶ Posets
Mp00206: Posets ā€”antichains of maximal sizeāŸ¶ Lattices
St001622: Lattices āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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
[1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => ([(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
[1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => ([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 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 $x,y\in L$.
The following 167 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001666The number of non-isomorphic subposets of a lattice which are lattices. St001820The size of the image of the pop stack sorting operator. St001623The number of doubly irreducible elements of a lattice. St001651The Frankl number of a lattice. St001846The number of elements which do not have a complement in the lattice. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000297The number of leading ones in a binary word. St000745The index of the last row whose first entry is the row number in a standard Young tableau. 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. St000439The position of the first down step of a Dyck path. St000069The number of maximal elements of a poset. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000823The number of unsplittable factors of the set partition. St000971The smallest closer of a set partition. St000909The number of maximal chains of maximal size in a poset. 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. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001809The index of the step at the first peak of maximal height in a Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000617The number of global maxima of a Dyck path. St000654The first descent of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000911The number of maximal antichains of maximal size in a poset. St000883The number of longest increasing subsequences of a permutation. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. St000141The maximum drop size of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000989The number of final rises of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001432The order dimension of the partition. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000738The first entry in the last row of a standard tableau. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000007The number of saliances of the permutation. St000308The height of the tree associated to a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000702The number of weak deficiencies of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001652The length of a longest interval of consecutive numbers. St000990The first ascent of a permutation. St001058The breadth of the ordered tree. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000335The difference of lower and upper interactions. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle 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. St000844The size of the largest block in the direct sum decomposition of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001530The depth of a Dyck path. St000051The size of the left subtree of a binary tree. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000209Maximum difference of elements in cycles. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000956The maximal displacement of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000443The number of long tunnels of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001959The product of the heights of the peaks of a Dyck path. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St001480The number of simple summands of the module J^2/J^3. St000845The maximal number of elements covered by an element in a poset. St000663The number of right floats of a permutation. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000651The maximal size of a rise in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000317The cycle descent number of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001130The number of two successive successions in a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St000260The radius of a connected graph. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nāˆ’1}]$ by adding $c_0$ to $c_{nāˆ’1}$. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000028The number of stack-sorts needed to sort a permutation. St000359The number of occurrences of the pattern 23-1. St001330The hat guessing number of a graph. St000906The length of the shortest maximal chain in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000670The reversal length of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000454The largest eigenvalue of a graph if it is integral. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001778The largest greatest common divisor of an element and its image in a permutation. St000624The normalized sum of the minimal distances to a greater element. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001394The genus of a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000649The number of 3-excedences of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001115The number of even descents of a permutation. St000652The maximal difference between successive positions of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001662The length of the longest factor of consecutive numbers in a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers.