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Your data matches 46 different statistics following compositions of up to 3 maps.
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Matching statistic: St001645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 8
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 8
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 8
[1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 8
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
[2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 4
Description
The pebbling number of a connected graph.
Matching statistic: St001621
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(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)
=> ? = 4
[2,1,5,3,4] => ([(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)
=> ? = 4
[2,3,1,5,4] => ([(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)
=> ? = 4
[2,4,1,5,3] => ([(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)
=> ? = 8
[2,4,5,1,3] => ([(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)
=> ? = 16
[3,1,2,5,4] => ([(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)
=> ? = 4
[3,1,5,2,4] => ([(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)
=> ? = 8
[3,4,1,5,2] => ([(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)
=> ? = 16
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[3,5,1,2,4] => ([(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)
=> ? = 16
[4,1,5,2,3] => ([(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)
=> ? = 16
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => ([(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)
=> ? = 4
[1,3,2,6,4,5] => ([(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)
=> ? = 4
[1,3,4,2,6,5] => ([(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)
=> ? = 4
[1,3,5,2,6,4] => ([(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)
=> ? = 8
[1,3,5,6,2,4] => ([(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)
=> ? = 16
[1,4,2,3,6,5] => ([(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)
=> ? = 4
[1,4,2,6,3,5] => ([(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)
=> ? = 8
[1,4,5,2,6,3] => ([(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)
=> ? = 16
[1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[1,4,6,2,3,5] => ([(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)
=> ? = 16
[1,5,2,6,3,4] => ([(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)
=> ? = 16
[1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => ([(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)
=> ? = 4
[2,1,3,6,4,5] => ([(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)
=> ? = 4
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => ([(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)
=> ? = 3
[2,1,4,5,3,6] => ([(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)
=> ? = 4
[2,1,5,3,4,6] => ([(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)
=> ? = 4
[2,3,1,4,6,5] => ([(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)
=> ? = 4
[2,3,1,5,4,6] => ([(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)
=> ? = 4
[2,3,1,5,6,4] => ([(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)
=> ? = 4
[2,3,1,6,4,5] => ([(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)
=> ? = 4
[2,4,1,5,3,6] => ([(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)
=> ? = 8
[2,4,5,1,3,6] => ([(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)
=> ? = 16
[3,1,2,4,6,5] => ([(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)
=> ? = 4
[3,1,2,5,4,6] => ([(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)
=> ? = 4
[3,1,2,5,6,4] => ([(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)
=> ? = 4
[3,1,2,6,4,5] => ([(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)
=> ? = 4
[3,1,5,2,4,6] => ([(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)
=> ? = 8
[3,2,1,6,5,4] => ([(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)
=> ? = 6
[3,4,1,5,2,6] => ([(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)
=> ? = 16
[3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[3,5,1,2,4,6] => ([(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)
=> ? = 16
[4,1,5,2,3,6] => ([(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)
=> ? = 16
[4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[1,2,3,4,5,7,6] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,3,4,6,5,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3,5,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3,6,7,5] => ([(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)
=> ? = 4
[1,2,4,3,7,5,6] => ([(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)
=> ? = 4
[1,2,4,5,3,7,6] => ([(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)
=> ? = 4
[1,2,4,6,3,7,5] => ([(2,6),(3,5),(4,5),(4,6)],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)
=> ? = 8
[1,3,2,4,5,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
=> ([(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: St001624
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => ([(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)
=> ? = 4
[2,1,5,3,4] => ([(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)
=> ? = 4
[2,3,1,5,4] => ([(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)
=> ? = 4
[2,4,1,5,3] => ([(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)
=> ? = 8
[2,4,5,1,3] => ([(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)
=> ? = 16
[3,1,2,5,4] => ([(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)
=> ? = 4
[3,1,5,2,4] => ([(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)
=> ? = 8
[3,4,1,5,2] => ([(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)
=> ? = 16
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[3,5,1,2,4] => ([(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)
=> ? = 16
[4,1,5,2,3] => ([(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)
=> ? = 16
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[1,2,3,4,6,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => ([(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)
=> ? = 4
[1,3,2,6,4,5] => ([(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)
=> ? = 4
[1,3,4,2,6,5] => ([(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)
=> ? = 4
[1,3,5,2,6,4] => ([(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)
=> ? = 8
[1,3,5,6,2,4] => ([(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)
=> ? = 16
[1,4,2,3,6,5] => ([(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)
=> ? = 4
[1,4,2,6,3,5] => ([(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)
=> ? = 8
[1,4,5,2,6,3] => ([(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)
=> ? = 16
[1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[1,4,6,2,3,5] => ([(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)
=> ? = 16
[1,5,2,6,3,4] => ([(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)
=> ? = 16
[1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[2,1,3,4,5,6] => ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => ([(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)
=> ? = 4
[2,1,3,6,4,5] => ([(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)
=> ? = 4
[2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => ([(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)
=> ? = 3
[2,1,4,5,3,6] => ([(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)
=> ? = 4
[2,1,5,3,4,6] => ([(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)
=> ? = 4
[2,3,1,4,6,5] => ([(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)
=> ? = 4
[2,3,1,5,4,6] => ([(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)
=> ? = 4
[2,3,1,5,6,4] => ([(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)
=> ? = 4
[2,3,1,6,4,5] => ([(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)
=> ? = 4
[2,4,1,5,3,6] => ([(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)
=> ? = 8
[2,4,5,1,3,6] => ([(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)
=> ? = 16
[3,1,2,4,6,5] => ([(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)
=> ? = 4
[3,1,2,5,4,6] => ([(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)
=> ? = 4
[3,1,2,5,6,4] => ([(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)
=> ? = 4
[3,1,2,6,4,5] => ([(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)
=> ? = 4
[3,1,5,2,4,6] => ([(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)
=> ? = 8
[3,2,1,6,5,4] => ([(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)
=> ? = 6
[3,4,1,5,2,6] => ([(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)
=> ? = 16
[3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[3,5,1,2,4,6] => ([(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)
=> ? = 16
[4,1,5,2,3,6] => ([(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)
=> ? = 16
[4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 8
[1,2,3,4,5,7,6] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,3,4,6,5,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3,5,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3,6,7,5] => ([(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)
=> ? = 4
[1,2,4,3,7,5,6] => ([(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)
=> ? = 4
[1,2,4,5,3,7,6] => ([(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)
=> ? = 4
[1,2,4,6,3,7,5] => ([(2,6),(3,5),(4,5),(4,6)],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)
=> ? = 8
[1,3,2,4,5,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5,6,7] => ([(5,6)],7)
=> ([(0,1)],2)
=> 1
[2,1,3,4,5,7,6] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,6,5,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 25%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 25%
Values
[2,1] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 4
[2,1,5,3,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 4
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 4
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[2,4,5,1,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ? = 16
[3,1,2,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
=> ? = 4
[3,1,5,2,4] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ? = 8
[3,4,1,5,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ? = 16
[3,4,5,1,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ? = 8
[3,5,1,2,4] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ? = 16
[4,1,5,2,3] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 16
[4,5,1,2,3] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ? = 8
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,3,2,5,6,4] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 4
[1,3,2,6,4,5] => [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 4
[1,3,4,2,6,5] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 4
[1,3,5,2,6,4] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[1,3,5,6,2,4] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ? = 16
[1,4,2,3,6,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
=> ? = 4
[1,4,2,6,3,5] => [1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ? = 8
[1,4,5,2,6,3] => [1,5,6,4,2,3] => ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ? = 16
[1,4,5,6,2,3] => [1,3,6,4,5,2] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ? = 8
[1,4,6,2,3,5] => [1,3,5,4,6,2] => ([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ? = 16
[1,5,2,6,3,4] => [1,5,4,6,2,3] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 16
[1,5,6,2,3,4] => [1,3,4,6,5,2] => ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ? = 8
[2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 2
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[2,1,3,5,6,4] => [2,1,3,6,5,4] => ([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 4
[2,1,3,6,4,5] => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 4
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 2
[2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(8,1),(8,2),(9,3),(9,4)],10)
=> ? = 3
[2,1,4,5,3,6] => [2,1,5,4,3,6] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 4
[2,1,5,3,4,6] => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 4
[2,3,1,4,6,5] => [3,2,1,4,6,5] => ([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 4
[2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 4
[2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,8),(5,8),(6,8),(8,1),(8,2),(8,3)],9)
=> ? = 4
[2,3,1,6,4,5] => [3,2,1,5,6,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,8),(4,8),(5,8),(6,1),(8,2),(8,6)],9)
=> ? = 4
[2,4,1,5,3,6] => [4,2,5,1,3,6] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 8
[2,4,5,1,3,6] => [3,2,5,4,1,6] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ? = 16
[3,1,2,4,6,5] => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
=> ? = 4
[3,1,2,5,4,6] => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
=> ? = 4
[3,1,2,5,6,4] => [2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(1,7),(2,8),(3,8),(4,8),(5,7),(6,1),(7,2),(7,3),(7,4)],9)
=> ? = 4
[3,1,2,6,4,5] => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,4),(0,6),(1,8),(2,7),(3,7),(4,8),(5,3),(6,1),(8,2),(8,5)],9)
=> ? = 4
[3,1,5,2,4,6] => [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> ? = 8
[3,2,1,6,5,4] => [3,1,2,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,4),(0,6),(1,8),(2,7),(3,7),(4,8),(5,3),(6,1),(8,2),(8,5)],9)
=> ? = 6
[3,4,1,5,2,6] => [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)
=> ? = 16
[3,4,5,1,2,6] => [2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ? = 8
[3,5,1,2,4,6] => [2,4,3,5,1,6] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ? = 16
[4,1,5,2,3,6] => [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 16
[4,5,1,2,3,6] => [2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ? = 8
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ([(0,6),(1,7),(2,7),(3,8),(4,8),(5,8),(6,1),(6,2),(7,3),(7,4),(7,5)],9)
=> ? = 4
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,4),(6,1),(6,2),(8,3),(8,5)],9)
=> ? = 4
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
=> ([(0,6),(1,8),(2,8),(3,8),(4,7),(5,7),(6,1),(6,2),(6,3),(8,4),(8,5)],9)
=> ? = 4
[1,2,4,6,3,7,5] => [1,2,6,4,7,3,5] => ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
=> ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3),(5,7),(6,7)],8)
=> ? = 8
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 2
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 2
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 1
[2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 2
[2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 2
[2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 2
[2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> 2
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000080
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000080: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000080: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,4,4],[4,4,6],[5,6],[6]]
=> ?
=> ? = 8
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 8
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,3,4,6],[4,4,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,5,5],[3,3,5,6],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,4,4,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 16
[1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,3,3],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,4,4,4],[3,4,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[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)
=> ? = 3
[2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,4,1,5,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,3,3],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[3,1,2,5,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[3,1,5,2,4,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(0,7),(0,8),(1,10),(1,45),(1,46),(2,9),(2,43),(2,44),(3,12),(3,48),(3,50),(4,11),(4,47),(4,49),(5,2),(5,39),(5,40),(6,1),(6,41),(6,42),(7,4),(7,39),(7,41),(8,3),(8,40),(8,42),(9,14),(9,19),(9,20),(10,14),(10,21),(10,22),(11,13),(11,23),(11,25),(12,13),(12,24),(12,26),(13,53),(13,54),(14,51),(14,52),(15,55),(15,57),(16,56),(16,57),(17,55),(17,58),(18,56),(18,58),(19,27),(19,51),(20,28),(20,52),(21,29),(21,51),(22,30),(22,52),(23,31),(23,53),(24,32),(24,53),(25,33),(25,54),(26,34),(26,54),(27,15),(27,59),(28,16),(28,60),(29,17),(29,59),(30,18),(30,60),(31,15),(31,61),(32,16),(32,61),(33,17),(33,62),(34,18),(34,62),(35,27),(35,31),(36,28),(36,32),(37,29),(37,33),(38,30),(38,34),(39,43),(39,47),(40,44),(40,48),(41,45),(41,49),(42,46),(42,50),(43,19),(43,35),(44,20),(44,36),(45,21),(45,37),(46,22),(46,38),(47,23),(47,35),(48,24),(48,36),(49,25),(49,37),(50,26),(50,38),(51,59),(52,60),(53,61),(54,62),(55,63),(56,63),(57,63),(58,63),(59,55),(60,56),(61,57),(62,58)],64)
=> ? = 6
[3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,2,3,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,4,5,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,4,4],[2,2,2,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,3,3,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,1,5,2,3,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,2,2,2],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,3,3,3],[2,2,3,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,5],[6,7],[7]]
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,6],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,7],[7]]
=> ?
=> ? = 2
Description
The rank of the poset.
Matching statistic: St000298
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,4,4],[4,4,6],[5,6],[6]]
=> ?
=> ? = 8
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 8
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,3,4,6],[4,4,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,5,5],[3,3,5,6],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,4,4,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 16
[1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,3,3],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,4,4,4],[3,4,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[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)
=> ? = 3
[2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,4,1,5,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,3,3],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[3,1,2,5,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[3,1,5,2,4,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(0,7),(0,8),(1,10),(1,45),(1,46),(2,9),(2,43),(2,44),(3,12),(3,48),(3,50),(4,11),(4,47),(4,49),(5,2),(5,39),(5,40),(6,1),(6,41),(6,42),(7,4),(7,39),(7,41),(8,3),(8,40),(8,42),(9,14),(9,19),(9,20),(10,14),(10,21),(10,22),(11,13),(11,23),(11,25),(12,13),(12,24),(12,26),(13,53),(13,54),(14,51),(14,52),(15,55),(15,57),(16,56),(16,57),(17,55),(17,58),(18,56),(18,58),(19,27),(19,51),(20,28),(20,52),(21,29),(21,51),(22,30),(22,52),(23,31),(23,53),(24,32),(24,53),(25,33),(25,54),(26,34),(26,54),(27,15),(27,59),(28,16),(28,60),(29,17),(29,59),(30,18),(30,60),(31,15),(31,61),(32,16),(32,61),(33,17),(33,62),(34,18),(34,62),(35,27),(35,31),(36,28),(36,32),(37,29),(37,33),(38,30),(38,34),(39,43),(39,47),(40,44),(40,48),(41,45),(41,49),(42,46),(42,50),(43,19),(43,35),(44,20),(44,36),(45,21),(45,37),(46,22),(46,38),(47,23),(47,35),(48,24),(48,36),(49,25),(49,37),(50,26),(50,38),(51,59),(52,60),(53,61),(54,62),(55,63),(56,63),(57,63),(58,63),(59,55),(60,56),(61,57),(62,58)],64)
=> ? = 6
[3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,2,3,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,4,5,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,4,4],[2,2,2,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,3,3,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,1,5,2,3,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,2,2,2],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,3,3,3],[2,2,3,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,5],[6,7],[7]]
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,6],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,7],[7]]
=> ?
=> ? = 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000307: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000307: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,4,4],[4,4,6],[5,6],[6]]
=> ?
=> ? = 8
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 8
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,3,4,6],[4,4,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,5,5],[3,3,5,6],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,4,4,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 16
[1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,3,3],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,4,4,4],[3,4,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[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)
=> ? = 3
[2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,4,1,5,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,3,3],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[3,1,2,5,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[3,1,5,2,4,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(0,7),(0,8),(1,10),(1,45),(1,46),(2,9),(2,43),(2,44),(3,12),(3,48),(3,50),(4,11),(4,47),(4,49),(5,2),(5,39),(5,40),(6,1),(6,41),(6,42),(7,4),(7,39),(7,41),(8,3),(8,40),(8,42),(9,14),(9,19),(9,20),(10,14),(10,21),(10,22),(11,13),(11,23),(11,25),(12,13),(12,24),(12,26),(13,53),(13,54),(14,51),(14,52),(15,55),(15,57),(16,56),(16,57),(17,55),(17,58),(18,56),(18,58),(19,27),(19,51),(20,28),(20,52),(21,29),(21,51),(22,30),(22,52),(23,31),(23,53),(24,32),(24,53),(25,33),(25,54),(26,34),(26,54),(27,15),(27,59),(28,16),(28,60),(29,17),(29,59),(30,18),(30,60),(31,15),(31,61),(32,16),(32,61),(33,17),(33,62),(34,18),(34,62),(35,27),(35,31),(36,28),(36,32),(37,29),(37,33),(38,30),(38,34),(39,43),(39,47),(40,44),(40,48),(41,45),(41,49),(42,46),(42,50),(43,19),(43,35),(44,20),(44,36),(45,21),(45,37),(46,22),(46,38),(47,23),(47,35),(48,24),(48,36),(49,25),(49,37),(50,26),(50,38),(51,59),(52,60),(53,61),(54,62),(55,63),(56,63),(57,63),(58,63),(59,55),(60,56),(61,57),(62,58)],64)
=> ? = 6
[3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,2,3,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,4,5,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,4,4],[2,2,2,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,3,3,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,1,5,2,3,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,2,2,2],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,3,3,3],[2,2,3,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,5],[6,7],[7]]
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,6],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,7],[7]]
=> ?
=> ? = 2
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000633
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000633: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000633: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,4,4],[4,4,6],[5,6],[6]]
=> ?
=> ? = 8
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 8
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,3,4,6],[4,4,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,5,5],[3,3,5,6],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,4,4,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 16
[1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,3,3],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,4,4,4],[3,4,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[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)
=> ? = 3
[2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,4,1,5,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,3,3],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[3,1,2,5,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[3,1,5,2,4,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(0,7),(0,8),(1,10),(1,45),(1,46),(2,9),(2,43),(2,44),(3,12),(3,48),(3,50),(4,11),(4,47),(4,49),(5,2),(5,39),(5,40),(6,1),(6,41),(6,42),(7,4),(7,39),(7,41),(8,3),(8,40),(8,42),(9,14),(9,19),(9,20),(10,14),(10,21),(10,22),(11,13),(11,23),(11,25),(12,13),(12,24),(12,26),(13,53),(13,54),(14,51),(14,52),(15,55),(15,57),(16,56),(16,57),(17,55),(17,58),(18,56),(18,58),(19,27),(19,51),(20,28),(20,52),(21,29),(21,51),(22,30),(22,52),(23,31),(23,53),(24,32),(24,53),(25,33),(25,54),(26,34),(26,54),(27,15),(27,59),(28,16),(28,60),(29,17),(29,59),(30,18),(30,60),(31,15),(31,61),(32,16),(32,61),(33,17),(33,62),(34,18),(34,62),(35,27),(35,31),(36,28),(36,32),(37,29),(37,33),(38,30),(38,34),(39,43),(39,47),(40,44),(40,48),(41,45),(41,49),(42,46),(42,50),(43,19),(43,35),(44,20),(44,36),(45,21),(45,37),(46,22),(46,38),(47,23),(47,35),(48,24),(48,36),(49,25),(49,37),(50,26),(50,38),(51,59),(52,60),(53,61),(54,62),(55,63),(56,63),(57,63),(58,63),(59,55),(60,56),(61,57),(62,58)],64)
=> ? = 6
[3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,2,3,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,4,5,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,4,4],[2,2,2,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,3,3,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,1,5,2,3,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,2,2,2],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,3,3,3],[2,2,3,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,5],[6,7],[7]]
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,6],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,7],[7]]
=> ?
=> ? = 2
Description
The size of the automorphism group of a poset.
A poset automorphism is a permutation of the elements of the poset preserving the order relation.
Matching statistic: St000640
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000640: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000640: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,4,4],[4,4,6],[5,6],[6]]
=> ?
=> ? = 8
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 8
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,3,4,6],[4,4,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,5,5],[3,3,5,6],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,4,4,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 16
[1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,3,3],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,4,4,4],[3,4,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[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)
=> ? = 3
[2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,4,1,5,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,3,3],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[3,1,2,5,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[3,1,5,2,4,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(0,7),(0,8),(1,10),(1,45),(1,46),(2,9),(2,43),(2,44),(3,12),(3,48),(3,50),(4,11),(4,47),(4,49),(5,2),(5,39),(5,40),(6,1),(6,41),(6,42),(7,4),(7,39),(7,41),(8,3),(8,40),(8,42),(9,14),(9,19),(9,20),(10,14),(10,21),(10,22),(11,13),(11,23),(11,25),(12,13),(12,24),(12,26),(13,53),(13,54),(14,51),(14,52),(15,55),(15,57),(16,56),(16,57),(17,55),(17,58),(18,56),(18,58),(19,27),(19,51),(20,28),(20,52),(21,29),(21,51),(22,30),(22,52),(23,31),(23,53),(24,32),(24,53),(25,33),(25,54),(26,34),(26,54),(27,15),(27,59),(28,16),(28,60),(29,17),(29,59),(30,18),(30,60),(31,15),(31,61),(32,16),(32,61),(33,17),(33,62),(34,18),(34,62),(35,27),(35,31),(36,28),(36,32),(37,29),(37,33),(38,30),(38,34),(39,43),(39,47),(40,44),(40,48),(41,45),(41,49),(42,46),(42,50),(43,19),(43,35),(44,20),(44,36),(45,21),(45,37),(46,22),(46,38),(47,23),(47,35),(48,24),(48,36),(49,25),(49,37),(50,26),(50,38),(51,59),(52,60),(53,61),(54,62),(55,63),(56,63),(57,63),(58,63),(59,55),(60,56),(61,57),(62,58)],64)
=> ? = 6
[3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,2,3,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,4,5,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,4,4],[2,2,2,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,3,3,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,1,5,2,3,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,2,2,2],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,3,3,3],[2,2,3,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,5],[6,7],[7]]
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,6],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,7],[7]]
=> ?
=> ? = 2
Description
The rank of the largest boolean interval in a poset.
Matching statistic: St000845
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000845: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000845: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 25%
Values
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? = 8
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? = 16
[4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,2],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? = 16
[4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? = 8
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,2,6,4,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[1,3,5,2,6,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,4,4],[4,4,6],[5,6],[6]]
=> ?
=> ? = 8
[1,3,5,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[1,4,2,6,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,3,3],[3,3,3,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 8
[1,4,5,2,6,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,3,4,6],[4,4,6],[5,6],[6]]
=> ?
=> ? = 16
[1,4,5,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,5,5],[3,3,5,6],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[1,4,6,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,4,4],[3,4,4,5],[4,5,5],[5,6],[6]]
=> ?
=> ? = 16
[1,5,2,6,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,3,3,3],[3,3,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 16
[1,5,6,2,3,4] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[1,1,1,1,1,1],[2,2,4,4,4],[3,4,5,5],[4,5,6],[5,6],[6]]
=> ?
=> ? = 8
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,3,6,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[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)
=> ? = 3
[2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,1,5,3,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4
[2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 4
[2,3,1,6,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[2,4,1,5,3,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,3,3],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,1,2,4,6,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,5,4,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ?
=> ? = 4
[3,1,2,5,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ?
=> ? = 4
[3,1,2,6,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,2],[2,2,2,2,3],[3,3,3,3],[4,5,5],[5,6],[6]]
=> ?
=> ? = 4
[3,1,5,2,4,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,2],[2,2,2,2,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(0,7),(0,8),(1,10),(1,45),(1,46),(2,9),(2,43),(2,44),(3,12),(3,48),(3,50),(4,11),(4,47),(4,49),(5,2),(5,39),(5,40),(6,1),(6,41),(6,42),(7,4),(7,39),(7,41),(8,3),(8,40),(8,42),(9,14),(9,19),(9,20),(10,14),(10,21),(10,22),(11,13),(11,23),(11,25),(12,13),(12,24),(12,26),(13,53),(13,54),(14,51),(14,52),(15,55),(15,57),(16,56),(16,57),(17,55),(17,58),(18,56),(18,58),(19,27),(19,51),(20,28),(20,52),(21,29),(21,51),(22,30),(22,52),(23,31),(23,53),(24,32),(24,53),(25,33),(25,54),(26,34),(26,54),(27,15),(27,59),(28,16),(28,60),(29,17),(29,59),(30,18),(30,60),(31,15),(31,61),(32,16),(32,61),(33,17),(33,62),(34,18),(34,62),(35,27),(35,31),(36,28),(36,32),(37,29),(37,33),(38,30),(38,34),(39,43),(39,47),(40,44),(40,48),(41,45),(41,49),(42,46),(42,50),(43,19),(43,35),(44,20),(44,36),(45,21),(45,37),(46,22),(46,38),(47,23),(47,35),(48,24),(48,36),(49,25),(49,37),(50,26),(50,38),(51,59),(52,60),(53,61),(54,62),(55,63),(56,63),(57,63),(58,63),(59,55),(60,56),(61,57),(62,58)],64)
=> ? = 6
[3,4,1,5,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,2,3,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[3,4,5,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,4,4],[2,2,2,4,5],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,3,3],[2,2,3,3,4],[3,3,4,4],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,1,5,2,3,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,2,2,2],[2,2,2,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 16
[4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,3,3,3],[2,2,3,4,4],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ?
=> ? = 8
[1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,5],[6,7],[7]]
=> ?
=> ? = 1
[1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,4],[5,5,6],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,6],[7]]
=> ?
=> ? = 1
[1,2,3,5,4,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [[1,1,1,1,1,1,1],[2,2,2,2,2,2],[3,3,3,3,3],[4,4,4,5],[5,5,5],[6,7],[7]]
=> ?
=> ? = 2
Description
The maximal number of elements covered by an element in a poset.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001779The order of promotion on the set of linear extensions of a poset. St000528The height of a poset. St000632The jump number of the poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St001545The second Elser number of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000422The energy of a graph, if it is integral. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2.
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