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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St001200
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,2)],4)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(2,4),(3,4)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(1,4),(2,3)],5)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(3,5),(4,5)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
([(2,5),(3,4)],6)
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001618
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ? = 3
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? = 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(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)
=> ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
Description
The cardinality of the Frattini sublattice of a lattice.
The Frattini sublattice is the intersection of all proper maximal sublattices of the lattice.
Matching statistic: St001720
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ? = 3
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? = 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(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)
=> ? = 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St001719
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ? = 3 - 1
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ? = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? = 3 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ? = 3 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ? = 3 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 1
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ? = 3 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(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)
=> ? = 3 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 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: St001820
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ? = 3 - 1
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ? = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? = 3 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ? = 3 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ? = 3 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 1
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ? = 3 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(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)
=> ? = 3 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 1 = 2 - 1
Description
The size of the image of the pop stack sorting operator.
The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001845
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ? = 3 - 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 2
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? = 3 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ? = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 2
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ? = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(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)
=> ? = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 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: St001846
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ? = 3 - 2
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ? = 3 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ? = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 2
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ? = 3 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ? = 3 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ? = 3 - 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 3 - 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ? = 3 - 2
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ? = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ? = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(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)
=> ? = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 2 - 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ? = 4 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 2 - 2
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Values
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => ? = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 10110 => ? = 2 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => ? = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 10110 => ? = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 3 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> 10110 => ? = 2 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [2,1,1,1]
=> 101110 => ? = 2 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> 10110 => ? = 2 - 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 3 - 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,2,2]
=> 101100 => ? = 3 - 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1]
=> [2,1,1,1]
=> 101110 => ? = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> [4,3]
=> [2,2,2,1]
=> 111010 => ? = 3 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,1]
=> [3,2,2]
=> 101100 => ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => ? = 3 - 1
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> [3,3,1]
=> [3,2,2]
=> 101100 => ? = 3 - 1
([(4,6),(5,6)],7)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(3,6),(4,5)],7)
=> [1,1]
=> [2]
=> 100 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1]
=> [2,1,1]
=> 10110 => ? = 2 - 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [3,1,1]
=> [3,1,1]
=> 100110 => ? = 3 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1]
=> [4]
=> 10000 => ? = 3 - 1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> [4,1]
=> [2,1,1,1]
=> 101110 => ? = 2 - 1
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> [1,1,1,1,1]
=> [5]
=> 100000 => ? = 4 - 1
([(1,6),(2,5),(3,4)],7)
=> [1,1,1]
=> [3]
=> 1000 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001621
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ?
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ?
=> ? = 3
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ?
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ?
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ?
=> ? = 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ?
=> ? = 3
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001623
Values
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ?
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ?
=> ? = 3
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ?
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,11),(1,15),(1,19),(1,23),(1,24),(1,25),(2,10),(2,14),(2,18),(2,21),(2,22),(2,25),(3,9),(3,13),(3,17),(3,20),(3,22),(3,24),(4,8),(4,12),(4,16),(4,20),(4,21),(4,23),(5,16),(5,17),(5,18),(5,19),(5,26),(6,12),(6,13),(6,14),(6,15),(6,26),(7,8),(7,9),(7,10),(7,11),(7,26),(8,27),(8,28),(8,30),(8,45),(9,27),(9,29),(9,31),(9,46),(10,28),(10,29),(10,32),(10,47),(11,30),(11,31),(11,32),(11,48),(12,33),(12,34),(12,36),(12,45),(13,33),(13,35),(13,37),(13,46),(14,34),(14,35),(14,38),(14,47),(15,36),(15,37),(15,38),(15,48),(16,39),(16,40),(16,42),(16,45),(17,39),(17,41),(17,43),(17,46),(18,40),(18,41),(18,44),(18,47),(19,42),(19,43),(19,44),(19,48),(20,27),(20,33),(20,39),(20,55),(21,28),(21,34),(21,40),(21,55),(22,29),(22,35),(22,41),(22,55),(23,30),(23,36),(23,42),(23,55),(24,31),(24,37),(24,43),(24,55),(25,32),(25,38),(25,44),(25,55),(26,45),(26,46),(26,47),(26,48),(27,51),(27,56),(28,49),(28,56),(29,50),(29,56),(30,52),(30,56),(31,53),(31,56),(32,54),(32,56),(33,51),(33,57),(34,49),(34,57),(35,50),(35,57),(36,52),(36,57),(37,53),(37,57),(38,54),(38,57),(39,51),(39,58),(40,49),(40,58),(41,50),(41,58),(42,52),(42,58),(43,53),(43,58),(44,54),(44,58),(45,49),(45,51),(45,52),(46,50),(46,51),(46,53),(47,49),(47,50),(47,54),(48,52),(48,53),(48,54),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,56),(55,57),(55,58),(56,59),(57,59),(58,59)],60)
=> ?
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ?
=> ? = 3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,20),(1,21),(1,22),(1,33),(2,9),(2,17),(2,18),(2,19),(2,33),(3,8),(3,14),(3,15),(3,16),(3,33),(4,13),(4,16),(4,19),(4,22),(4,32),(5,12),(5,15),(5,18),(5,21),(5,32),(6,11),(6,14),(6,17),(6,20),(6,32),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,23),(8,24),(8,25),(8,40),(9,26),(9,27),(9,28),(9,40),(10,29),(10,30),(10,31),(10,40),(11,23),(11,26),(11,29),(11,41),(12,24),(12,27),(12,30),(12,41),(13,25),(13,28),(13,31),(13,41),(14,23),(14,34),(14,37),(15,24),(15,35),(15,37),(16,25),(16,36),(16,37),(17,26),(17,34),(17,38),(18,27),(18,35),(18,38),(19,28),(19,36),(19,38),(20,29),(20,34),(20,39),(21,30),(21,35),(21,39),(22,31),(22,36),(22,39),(23,42),(23,45),(24,43),(24,45),(25,44),(25,45),(26,42),(26,46),(27,43),(27,46),(28,44),(28,46),(29,42),(29,47),(30,43),(30,47),(31,44),(31,47),(32,37),(32,38),(32,39),(32,41),(33,34),(33,35),(33,36),(33,40),(34,42),(34,48),(35,43),(35,48),(36,44),(36,48),(37,45),(37,48),(38,46),(38,48),(39,47),(39,48),(40,42),(40,43),(40,44),(41,45),(41,46),(41,47),(42,49),(43,49),(44,49),(45,49),(46,49),(47,49),(48,49)],50)
=> ?
=> ? = 3
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(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)
=> ([(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)
=> ? = 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> ? = 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> ?
=> ? = 4
([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The number of doubly irreducible elements of a lattice.
An element $d$ of a lattice $L$ is '''doubly irreducible''' if it is both join and meet irreducible. That means, $d$ is neither the least nor the greatest element of $L$ and if $d=x\vee y$ or $d=x\wedge y$, then $d\in\{x,y\}$ for all $x,y\in L$.
In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
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