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
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
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
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
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
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
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
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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.
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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.
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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.
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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.
Mp00047: Ordered trees to posetPosets
Mp00206: Posets antichains of maximal sizeLattices
Mp00193: Lattices to posetPosets
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.