- FindStat

Your data matches 20 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001568
(load all 3 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St000929

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => []
 => ? = 1 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => []
 => ? = 1 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 2 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2]
 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => []
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => []
 => ? = 1 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1 = 2 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 2 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => []
 => ? = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1]
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [4,3]
 => [3]
 => []
 => ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 0 = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [5,2]
 => [2]
 => []
 => ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => []
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [4,3]
 => [3]
 => []
 => ? = 1 - 1
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,2,2]
 => [2,2]
 => [2]
 => 0 = 1 - 1
([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,7),(4,7),(5,7),(6,7)],8)
 => [5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([],8)
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1 = 2 - 1
([(4,7),(5,6)],8)
 => [2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 2 - 1
([(4,7),(5,6),(6,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(4,6),(4,7),(5,6),(5,7)],8)
 => [4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(2,7),(3,7),(4,6),(5,6)],8)
 => [3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
 => [5,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
 => [3,3,1,1]
 => [3,1,1]
 => [1,1]
 => 1 = 2 - 1
([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1 = 2 - 1
([(0,7),(1,6),(2,5),(3,4)],8)
 => [2,2,2,2]
 => [2,2,2]
 => [2,2]
 => 0 = 1 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)
 => [4,2,2]
 => [2,2]
 => [2]
 => 0 = 1 - 1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [2,2]
 => [2]
 => 0 = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => [4,2,2]
 => [2,2]
 => [2]
 => 0 = 1 - 1
([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
 => [6,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => [6,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => [9,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => [11,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 2 - 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => [7,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
 => [11,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
 => [13,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => [8,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
 => [12,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1 = 2 - 1

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Matching statistic: St001613
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001613: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 1 - 1
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 2 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1

Description

The binary logarithm of the size of the center of a lattice. An element of a lattice is central if it is neutral and has a complement. The subposet induced by central elements is a Boolean lattice.

Matching statistic: St001719
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 1 - 1
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 2 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

Matching statistic: St001881
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001881: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 1 - 1
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 2 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 1 = 2 - 1

Description

The number of factors of a lattice as a Cartesian product of lattices. Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

Matching statistic: St001845
(load all 3 compositions to match this statistic)

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001845: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

([],3)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([],4)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(0,3),(1,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([],5)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ?
 => ? = 1 - 2
([],6)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 2
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ?
 => ? = 1 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ?
 => ? = 1 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([],7)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 2 - 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(3,8),(4,10),(4,11),(5,7),(5,10),(6,7),(6,11),(7,13),(8,12),(9,12),(10,3),(10,13),(11,2),(11,13),(12,1),(13,8),(13,9)],14)
 => ? = 2 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 0 = 2 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ?
 => ? = 1 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ?
 => ? = 1 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ?
 => ? = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ?
 => ? = 1 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ?
 => ? = 1 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
 => ? = 2 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(2,9),(2,11),(3,8),(3,10),(4,12),(4,13),(5,3),(5,12),(5,14),(6,2),(6,13),(6,14),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18),(18,1)],19)
 => ? = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 1 - 2
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => 0 = 2 - 2

Description

The number of join irreducibles minus the rank of a lattice. A lattice is join-extremal, if this statistic is

$0$ .

Matching statistic: St001681

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001681: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

([],3)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([],4)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([],5)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ?
 => ? = 1 - 1
([],6)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 1
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ([(3,4),(3,5)],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ([],5)
 => ?
 => ? = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ([],5)
 => ?
 => ? = 1 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([],7)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 1
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 1
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? = 2 - 1
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
 => ?
 => ? = 1 - 1
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
 => ?
 => ? = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ?
 => ? = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
 => ?
 => ? = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ([(3,4),(3,5)],6)
 => ?
 => ? = 1 - 1
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 2 - 1
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2 - 1
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,5),(3,6),(3,13),(4,12),(5,9),(5,11),(6,9),(6,10),(7,15),(8,15),(9,14),(10,7),(10,14),(11,8),(11,14),(12,7),(12,8),(13,10),(13,11),(13,12),(14,15)],16)
 => ? = 2 - 1
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 1 - 1
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 1 = 2 - 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: St001677

Mapping

Mp00243: Graphs —weak duplicate order⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001677: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%

Values

([],3)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([],4)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(0,3),(1,2)],4)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([],5)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(2,4),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(1,4),(2,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],5)
 => ?
 => ? = 1 - 2
([],6)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(2,5),(3,5),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(2,5),(3,4)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 2
([(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 2
([(1,2),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2
([(0,5),(1,4),(2,3)],6)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ([(3,4),(3,5)],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([],5)
 => ([],5)
 => ?
 => ? = 1 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],5)
 => ([],5)
 => ?
 => ? = 1 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([],7)
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 0 = 2 - 2
([(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(3,6),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 2
([(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 - 2
([(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => ([],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
 => ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
 => ? = 2 - 2
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 2 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 0 = 2 - 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
 => ?
 => ? = 1 - 2
([(2,6),(3,5),(4,5),(4,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ?
 => ? = 2 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(5,2),(6,1)],7)
 => ?
 => ? = 1 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => ([],6)
 => ([],6)
 => ?
 => ? = 1 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
 => ?
 => ? = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(6,1),(6,2)],7)
 => ?
 => ? = 1 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,5),(4,5)],6)
 => ([(3,4),(3,5)],6)
 => ?
 => ? = 1 - 2
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 2 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 2 - 2
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2 - 2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
 => ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,5),(3,6),(3,13),(4,12),(5,9),(5,11),(6,9),(6,10),(7,15),(8,15),(9,14),(10,7),(10,14),(11,8),(11,14),(12,7),(12,8),(13,10),(13,11),(13,12),(14,15)],16)
 => ? = 2 - 2
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
 => ([(0,1),(1,2),(1,3),(1,4),(2,12),(2,13),(3,6),(3,13),(3,14),(4,5),(4,12),(4,14),(5,8),(5,10),(6,9),(6,11),(7,18),(8,16),(9,17),(10,7),(10,16),(11,7),(11,17),(12,8),(12,15),(13,9),(13,15),(14,10),(14,11),(14,15),(15,16),(15,17),(16,18),(17,18)],19)
 => ? = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ?
 => ? = 1 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ?
 => ? = 1 - 2
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => 0 = 2 - 2

Description

The number of non-degenerate subsets of a lattice whose meet is the bottom element. A subset whose meet is the bottom element is non-degenerate, if it neither contains the bottom, nor the top element of the lattice.

Matching statistic: St000284
(load all 2 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%

Values

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

Description

The Plancherel distribution on integer partitions. This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions. Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.

Matching statistic: St000698
(load all 2 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. For any positive integer

$k$ , one associates a

$k$ -core to a partition by repeatedly removing all rim hooks of size

$k$ . This statistic counts the

$2$ -rim hooks that are removed in this process to obtain a

$2$ -core.

The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.