- FindStat

Your data matches 47 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001876

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: 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)
 => 0
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]]],[[]]]
 => ([(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)
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[]]],[[]]]
 => ([(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)
 => 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]]],[],[[]]]
 => ([(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)
 => 0
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0

Description

The number of 2-regular simple modules in the incidence algebra of the lattice.

Matching statistic: St001877

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: 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)
 => 0
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]]],[[]]]
 => ([(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)
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[]]],[[]]]
 => ([(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)
 => 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 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
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]]],[],[[]]]
 => ([(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)
 => 0
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 0

Description

Number of indecomposable injective modules with projective dimension 2.

Matching statistic: St001681

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001681: 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 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 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)
 => 3 = 2 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 1
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 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)
 => 3 = 2 + 1
[[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 1
[[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 1
[[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 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)
 => 3 = 2 + 1
[[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 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)
 => 3 = 2 + 1
[[[]],[[],[[]]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[]],[[[]],[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 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)
 => 3 = 2 + 1
[[[],[]],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[],[[]]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]],[[]]]
 => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1

Description

The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. For example, the pentagon lattice has three such sets: the bottom element, and the two antichains of size two. The cube is the smallest lattice which has such sets of three different sizes: the bottom element, six antichains of size two and one antichain of size three.

Matching statistic: St001633

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001633: 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)
 => 0
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[]],[[]]]
 => ([(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)
 => 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[]]]]]
 => ([(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)
 => 0
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[]],[[]]]
 => ([(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)
 => 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[[]]]]]
 => ([(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)
 => 0
[[[]],[],[[]]]
 => ([(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)
 => 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)
 => 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),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]]],[[]]]
 => ([(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)
 => 0
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]],[[]]]]
 => ([(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)
 => 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[[]]]]]]
 => ([(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)
 => 0
[[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[[]],[[]]]
 => ([(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)
 => 1
[[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[],[[[[]]]]]
 => ([(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)
 => 0
[[],[[]],[],[[]]]
 => ([(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)
 => 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)
 => 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),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[]]],[[]]]
 => ([(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)
 => 0
[[],[[],[[[]]]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[]],[[]]]]
 => ([(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)
 => 1
[[],[[[[]]],[]]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[],[[[[[]]]]]]
 => ([(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)
 => 0
[[[]],[],[],[[]]]
 => ([(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)
 => 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)
 => 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),(1,6),(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)
 => 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
[[[]],[[],[[]]]]
 => ([(0,5),(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)
 => 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)
 => 1
[[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[[]]],[],[[]]]
 => ([(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)
 => 0
[[[[]]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[[[[]]]],[],[]]
 => ([(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)
 => 0
[[[],[[]]],[[]]]
 => ([(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)
 => 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)
 => 1
[[[],[[[]]]],[]]
 => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0

Description

The number of simple modules with projective dimension two in the incidence algebra of the poset.

Matching statistic: St000071

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000071: 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 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 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 + 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 + 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)
 => 3 = 2 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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 = 1 + 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 + 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)
 => 3 = 2 + 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)
 => 3 = 2 + 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 + 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 + 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 + 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 = 1 + 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 = 1 + 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 + 1

Description

The number of maximal chains in a poset.

Matching statistic: St000909

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000909: 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 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 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 + 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 + 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)
 => 3 = 2 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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 = 1 + 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 + 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)
 => 3 = 2 + 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)
 => 3 = 2 + 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 + 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 + 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 + 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 = 1 + 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 = 1 + 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 + 1

Description

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

Matching statistic: St000848
(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
St000848: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The balance constant multiplied with the number of linear extensions of a poset. A pair of elements

$x,y$ of a poset is

$\alpha$ -balanced if the proportion

$P(x,y)$ of linear extensions where

$x$ comes before

$y$ is between

$\alpha$ and

$1-\alpha$ . The balance constant of a poset is

$\max\min(P(x,y), P(y,x)).$ Kislitsyn [1] conjectured that every poset which is not a chain is

$1/3$ -balanced. Brightwell, Felsner and Trotter [2] show that it is at least

$(1-\sqrt 5)/10$ -balanced. Olson and Sagan [3] exhibit various posets that are

$1/2$ -balanced.

Matching statistic: St000849
(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
St000849: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of 1/3-balanced pairs in a poset. A pair of elements

$x,y$ of a poset is

$\alpha$ -balanced if the proportion of linear extensions where

$x$ comes before

$y$ is between

$\alpha$ and

$1-\alpha$ . Kislitsyn [1] conjectured that every poset which is not a chain has a

$1/3$ -balanced pair. Brightwell, Felsner and Trotter [2] show that at least a

$(1-\sqrt 5)/10$ -balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a

$1/3$ -balanced pair.

Matching statistic: St000910
(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
St000910: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 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 + 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 + 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)
 => 3 = 2 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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 = 1 + 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 + 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)
 => 3 = 2 + 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)
 => 3 = 2 + 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 + 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 + 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 + 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 = 1 + 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 = 1 + 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 + 1
[[[[[[[]]]]]]]
 => ([(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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0 + 1
[[],[[[[[[[]]]]]]]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ([(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)
 => ? = 0 + 1
[[[[[[[[]]]]]]],[]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ([(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)
 => ? = 0 + 1

Description

The number of maximal chains of minimal length in a poset.

Matching statistic: St001105
(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
St001105: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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 + 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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)
 => 3 = 2 + 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 + 1
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 = 1 + 1
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 + 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)
 => 3 = 2 + 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 + 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 + 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 = 1 + 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 + 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 + 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 = 1 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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)
 => 3 = 2 + 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 = 1 + 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 = 1 + 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 + 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)
 => 3 = 2 + 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)
 => 3 = 2 + 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 + 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 + 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 + 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 = 1 + 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 = 1 + 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 + 1
[[[[[[[]]]]]]]
 => ([(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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0 + 1
[[],[[[[[[[]]]]]]]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ([(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)
 => ? = 0 + 1
[[[[[[[[]]]]]]],[]]
 => ([(0,8),(1,7),(2,8),(3,4),(4,6),(5,3),(6,2),(7,5)],9)
 => ([(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)
 => ? = 0 + 1

Description

The number of greedy linear extensions of a poset. A linear extension of a poset

$P$ with elements

$\{x_1,\dots,x_n\}$ is greedy, if it can be obtained by the following algorithm: * Step 1. Choose a minimal element

$x_1$ . * Step 2. Suppose

$X=\{x_1,\dots,x_i\}$ have been chosen. If there is at least one minimal element of

$P\setminus X$ which is greater than

$x_i$ then choose

$x_{i+1}$ to be any such minimal element; otherwise, choose

$x_{i+1}$ to be any minimal element of

$P\setminus X$ . This statistic records the number of greedy linear extensions.

The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000100The number of linear extensions of a poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of 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". St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in 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. St001964The interval resolution global dimension of a poset. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000407The number of occurrences of the pattern 2143 in a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000570The Edelman-Greene number of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. 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.