Your data matches 73 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001613
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
Description
The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.
Matching statistic: St001621
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
Description
The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
Description
The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001878
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001881
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1
Description
The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.
Matching statistic: St000298
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
Description
The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000527
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
Description
The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000845
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
(load all 2 compositions to match this statistic)
Mapping
Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
Description
The maximal number of elements covering an element of a poset.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000665The number of rafts of a permutation. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000010The length of the partition. St000346The number of coarsenings of a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000481The number of upper covers of a partition in dominance order. St000147The largest part of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000480The number of lower covers of a partition in dominance order. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000353The number of inner valleys of a permutation. St000023The number of inner peaks of a permutation. St000779The tier of a permutation. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000099The number of valleys of a permutation, including the boundary. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001469The holeyness of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000092The number of outer peaks of a permutation. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000252The number of nodes of degree 3 of a binary tree. St000354The number of recoils of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000668The least common multiple of the parts of the partition. St000160The multiplicity of the smallest part of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000897The number of different multiplicities of parts of an integer partition. St001091The number of parts in an integer partition whose next smaller part has the same size. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001571The Cartan determinant of the integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001271The competition number of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.