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Your data matches 227 different statistics following compositions of up to 3 maps.
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Matching statistic: St000527
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(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 3
([(1,2)],3)
=> 2
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([],4)
=> 4
([(2,3)],4)
=> 3
([(1,2),(1,3)],4)
=> 3
([(0,1),(0,2),(0,3)],4)
=> 3
([(0,2),(0,3),(3,1)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> 2
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 2
([],5)
=> 5
([(3,4)],5)
=> 4
([(2,3),(2,4)],5)
=> 4
([(1,2),(1,3),(1,4)],5)
=> 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(4,2)],5)
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(2,3),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
([(2,4),(3,4)],5)
=> 4
([(1,4),(2,4),(4,3)],5)
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000010
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(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 87%●distinct values known / distinct values provided: 58%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 87%●distinct values known / distinct values provided: 58%
Values
([],1)
=> [1]
=> []
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [1]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ?
=> ?
=> ? = 9 - 1
([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ?
=> ?
=> ? = 10 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ?
=> ?
=> ? = 10 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(1,8),(1,18),(2,11),(2,13),(2,14),(2,18),(3,10),(3,12),(3,14),(3,18),(4,8),(4,9),(4,12),(4,13),(5,6),(5,9),(5,10),(5,11),(6,23),(8,19),(8,23),(9,16),(9,17),(9,23),(10,16),(10,20),(10,23),(11,17),(11,20),(11,23),(12,15),(12,16),(12,19),(13,15),(13,17),(13,19),(14,15),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21)],24)
=> ?
=> ?
=> ? = 9 - 1
([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ?
=> ? = 6 - 1
Description
The length of the partition.
Matching statistic: St000147
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 87%●distinct values known / distinct values provided: 58%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 87%●distinct values known / distinct values provided: 58%
Values
([],1)
=> [1]
=> []
=> []
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> []
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> [5,4,4,3,3,1,1]
=> ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> [5,4,4,4,3,2,2,1]
=> ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ?
=> ?
=> ?
=> ? = 10 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ?
=> ?
=> ?
=> ? = 10 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(1,8),(1,18),(2,11),(2,13),(2,14),(2,18),(3,10),(3,12),(3,14),(3,18),(4,8),(4,9),(4,12),(4,13),(5,6),(5,9),(5,10),(5,11),(6,23),(8,19),(8,23),(9,16),(9,17),(9,23),(10,16),(10,20),(10,23),(11,17),(11,20),(11,23),(12,15),(12,16),(12,19),(13,15),(13,17),(13,19),(14,15),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ?
=> ?
=> ? = 6 - 1
Description
The largest part of an integer partition.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 87%●distinct values known / distinct values provided: 58%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 87%●distinct values known / distinct values provided: 58%
Values
([],1)
=> [1]
=> []
=> []
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> []
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 2 = 3 - 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 5 - 1
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,3,1]
=> [9,2,1,1]
=> ? = 5 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> ?
=> ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> ?
=> ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ?
=> ?
=> ?
=> ? = 10 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ?
=> ?
=> ?
=> ? = 10 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,12),(2,13),(2,14),(2,18),(3,10),(3,11),(3,14),(3,18),(4,8),(4,9),(4,11),(4,13),(5,7),(5,9),(5,10),(5,12),(7,15),(7,19),(8,15),(8,20),(9,15),(9,16),(9,17),(10,16),(10,19),(10,23),(11,16),(11,20),(11,23),(12,17),(12,19),(12,23),(13,17),(13,20),(13,23),(14,23),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,6),(22,6),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(1,8),(1,18),(2,11),(2,13),(2,14),(2,18),(3,10),(3,12),(3,14),(3,18),(4,8),(4,9),(4,12),(4,13),(5,6),(5,9),(5,10),(5,11),(6,23),(8,19),(8,23),(9,16),(9,17),(9,23),(10,16),(10,20),(10,23),(11,17),(11,20),(11,23),(12,15),(12,16),(12,19),(13,15),(13,17),(13,19),(14,15),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,19),(18,20),(18,23),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000288
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [1]
=> []
=> => ? = 1 - 1
([],2)
=> [1,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> => ? = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> => ? = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 10 => 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 100 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> => ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> 100 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 100 => 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> 100 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 10 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> => ? = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> []
=> => ? = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> []
=> => ? = 1 - 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> []
=> => ? = 1 - 1
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> []
=> => ? = 1 - 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [7,5,1]
=> 1001000010 => ? = 4 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> 10010010010010 => ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> ? => ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> ? => ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ? => ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ? => ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ? => ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ? => ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ? => ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ? => ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ? => ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ? => ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ? => ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ? => ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ? => ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ? => ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ? => ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ? => ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ?
=> ?
=> ? => ? = 7 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000733
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1 - 1
([],2)
=> [1,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> []
=> []
=> ? = 1 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> []
=> []
=> ? = 1 - 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> []
=> []
=> ? = 1 - 1
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> []
=> []
=> ? = 1 - 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [7,5,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
=> ? = 4 - 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 5 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24],[25]]
=> ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> ?
=> ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> ?
=> ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001227
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 85%●distinct values known / distinct values provided: 42%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 85%●distinct values known / distinct values provided: 42%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1 - 1
([],2)
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? = 1 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> []
=> []
=> ? = 1 - 1
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 7 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> []
=> []
=> ? = 1 - 1
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> []
=> []
=> ? = 1 - 1
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> []
=> []
=> ? = 1 - 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [7,5,1]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 5 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> ?
=> ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> ?
=> ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000329
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1 - 2
([],2)
=> [1,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,1)],2)
=> [2]
=> []
=> []
=> ? = 1 - 2
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? = 1 - 2
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? = 1 - 2
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 5 - 2
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1,0]
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 3 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? = 1 - 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> []
=> []
=> ? = 1 - 2
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> []
=> []
=> ? = 1 - 2
([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> []
=> []
=> ? = 1 - 2
([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> [9]
=> []
=> []
=> ? = 1 - 2
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 4 - 2
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 - 2
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5 - 2
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 2
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 5 - 2
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 6 - 2
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 6 - 2
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [7,5,5,3,1]
=> ?
=> ? = 6 - 2
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [8,7,5,4,1]
=> ?
=> ? = 6 - 2
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ?
=> ? = 7 - 2
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ?
=> ? = 7 - 2
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ?
=> ? = 7 - 2
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ?
=> ? = 7 - 2
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 2
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ?
=> ? = 3 - 2
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 2
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ?
=> ? = 4 - 2
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ?
=> ? = 6 - 2
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5 - 2
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ?
=> ? = 6 - 2
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5 - 2
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 2
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ?
=> ? = 4 - 2
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ?
=> ? = 6 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 2
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7 - 2
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8 - 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 84%●distinct values known / distinct values provided: 58%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 84%●distinct values known / distinct values provided: 58%
Values
([],1)
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],2)
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1 = 2 - 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 3 - 1
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15],[16]]
=> ? = 4 - 1
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18],[19,20,21,22,23],[24]]
=> ? = 4 - 1
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24],[25]]
=> ? = 5 - 1
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15,16],[17,18,19],[20]]
=> ? = 5 - 1
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ? = 6 - 1
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18,19,20],[21,22,23,24,25,26,27],[28,29,30,31,32],[33,34,35],[36]]
=> ? = 6 - 1
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16],[17,18,19,20,21],[22,23,24,25,26],[27,28,29],[30]]
=> ? = 6 - 1
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12,13,14,15,16,17,18,19],[20,21,22,23,24,25,26],[27,28,29,30,31],[32,33,34,35],[36]]
=> ? = 6 - 1
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ? = 7 - 1
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ? = 7 - 1
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ? = 7 - 1
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ? = 7 - 1
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ? = 8 - 1
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ? = 8 - 1
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ? = 8 - 1
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14]]
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13,14],[15]]
=> ? = 6 - 1
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ? = 4 - 1
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ? = 9 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ? = 8 - 1
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ? = 8 - 1
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ?
=> ?
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ?
=> ?
=> ? = 9 - 1
([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ?
=> ?
=> ? = 7 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000734
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 84%●distinct values known / distinct values provided: 58%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 58% ●values known / values provided: 84%●distinct values known / distinct values provided: 58%
Values
([],1)
=> [1]
=> [1]
=> [[1]]
=> 1
([],2)
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
([(0,1)],2)
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
([],3)
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
([(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,1),(0,2)],3)
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([],4)
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(1,2),(1,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([],5)
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15],[16]]
=> ? = 4
([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
=> [11,7,5,1]
=> [4,3,3,3,3,2,2,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13],[14,15,16],[17,18],[19,20],[21],[22],[23],[24]]
=> ? = 4
([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13],[14,15,16],[17,18,19],[20,21],[22,23],[24],[25]]
=> ? = 5
([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
=> [7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13],[14,15,16],[17,18],[19],[20]]
=> ? = 5
([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15],[16,17,18],[19,20],[21]]
=> ? = 6
([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
=> [11,9,7,5,3,1]
=> [6,5,5,4,4,3,3,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15,16],[17,18,19,20],[21,22,23,24],[25,26,27],[28,29,30],[31,32],[33,34],[35],[36]]
=> ? = 6
([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
=> [9,7,5,5,3,1]
=> [6,5,5,4,4,2,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15,16],[17,18,19,20],[21,22,23,24],[25,26],[27,28],[29],[30]]
=> ? = 6
([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
=> [11,8,7,5,4,1]
=> [6,5,5,5,4,3,3,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15,16],[17,18,19,20,21],[22,23,24,25],[26,27,28],[29,30,31],[32,33],[34],[35],[36]]
=> ? = 6
([(0,17),(1,16),(2,26),(2,27),(3,24),(3,26),(4,25),(4,27),(5,16),(5,24),(6,17),(6,25),(8,10),(9,11),(10,12),(11,13),(12,14),(13,15),(14,7),(15,7),(16,8),(17,9),(18,21),(18,22),(19,10),(19,21),(20,11),(20,22),(21,12),(21,23),(22,13),(22,23),(23,14),(23,15),(24,8),(24,19),(25,9),(25,20),(26,18),(26,19),(27,18),(27,20)],28)
=> ?
=> ?
=> ?
=> ? = 7
([(0,28),(1,27),(2,28),(2,47),(3,45),(3,46),(4,46),(4,47),(5,45),(5,48),(6,27),(6,48),(8,25),(9,23),(10,24),(11,26),(12,13),(13,7),(14,21),(15,22),(16,18),(17,16),(18,20),(19,17),(20,15),(21,19),(22,13),(23,11),(23,35),(24,9),(24,34),(25,10),(25,33),(26,12),(26,22),(27,14),(28,8),(28,36),(29,30),(29,33),(30,31),(30,41),(31,34),(31,42),(32,30),(32,43),(33,24),(33,31),(34,23),(34,40),(35,15),(35,26),(36,25),(36,29),(37,29),(37,32),(38,32),(38,39),(39,19),(39,43),(40,20),(40,35),(41,16),(41,42),(42,18),(42,40),(43,17),(43,41),(44,21),(44,39),(45,38),(45,44),(46,37),(46,38),(47,36),(47,37),(48,14),(48,44)],49)
=> ?
=> ?
=> ?
=> ? = 7
([(0,34),(1,33),(2,27),(3,31),(3,37),(4,30),(4,31),(5,27),(5,30),(6,33),(6,34),(6,37),(8,26),(9,25),(10,14),(11,41),(12,41),(13,39),(14,7),(15,16),(16,14),(17,20),(17,39),(18,21),(18,39),(19,22),(20,23),(20,40),(21,24),(21,40),(22,29),(23,11),(23,38),(24,12),(24,38),(25,10),(25,16),(26,9),(26,32),(27,19),(28,22),(28,36),(29,11),(29,12),(30,19),(30,28),(31,28),(31,35),(32,15),(32,25),(33,13),(33,17),(34,13),(34,18),(35,20),(35,21),(35,36),(36,23),(36,24),(36,29),(37,17),(37,18),(37,35),(38,32),(38,41),(39,8),(39,40),(40,26),(40,38),(41,15)],42)
=> ?
=> ?
=> ?
=> ? = 7
([(0,49),(1,40),(2,41),(3,40),(3,46),(4,46),(4,52),(5,41),(5,51),(6,49),(6,51),(6,52),(7,9),(9,8),(10,48),(11,39),(12,24),(13,14),(13,47),(14,27),(15,22),(15,34),(16,23),(16,33),(17,60),(18,57),(19,56),(20,58),(21,13),(21,58),(22,11),(22,62),(23,10),(23,61),(24,7),(25,38),(26,24),(27,26),(28,17),(28,59),(29,28),(29,53),(30,42),(31,19),(31,53),(32,18),(32,60),(33,15),(33,35),(33,61),(34,21),(34,62),(35,22),(35,54),(36,37),(36,56),(37,18),(37,55),(38,12),(38,26),(39,25),(40,30),(41,45),(42,17),(42,32),(43,36),(43,44),(43,53),(44,32),(44,37),(44,59),(45,19),(45,36),(46,30),(46,50),(47,27),(47,38),(48,20),(48,21),(49,29),(49,31),(50,28),(50,42),(50,44),(51,31),(51,43),(51,45),(52,29),(52,43),(52,50),(53,16),(53,56),(53,59),(54,20),(54,62),(55,57),(55,61),(56,23),(56,55),(57,54),(58,25),(58,47),(59,33),(59,55),(59,60),(60,35),(60,57),(61,34),(61,48),(61,54),(62,39),(62,58)],63)
=> ?
=> ?
=> ?
=> ? = 7
([(0,20),(1,19),(2,31),(2,33),(3,32),(3,33),(4,31),(4,34),(5,32),(5,35),(6,19),(6,34),(7,20),(7,35),(9,15),(10,16),(11,17),(12,18),(13,11),(14,12),(15,13),(16,14),(17,8),(18,8),(19,9),(20,10),(21,22),(21,23),(22,11),(22,24),(23,12),(23,24),(24,17),(24,18),(25,21),(25,27),(26,21),(26,28),(27,13),(27,22),(28,14),(28,23),(29,15),(29,27),(30,16),(30,28),(31,25),(31,29),(32,26),(32,30),(33,25),(33,26),(34,9),(34,29),(35,10),(35,30)],36)
=> ?
=> ?
=> ?
=> ? = 8
([(0,33),(1,32),(2,33),(2,62),(3,60),(3,61),(4,59),(4,61),(5,60),(5,62),(6,59),(6,63),(7,32),(7,63),(9,30),(10,28),(11,27),(12,29),(13,31),(14,15),(15,8),(16,25),(17,26),(18,20),(19,18),(20,22),(21,19),(22,24),(23,21),(24,17),(25,23),(26,15),(27,13),(27,46),(28,11),(28,45),(29,10),(29,43),(30,12),(30,44),(31,14),(31,26),(32,16),(33,9),(33,47),(34,35),(34,40),(35,38),(35,43),(36,34),(36,39),(37,34),(37,44),(38,41),(38,53),(39,40),(39,55),(40,38),(40,54),(41,45),(41,56),(42,39),(42,57),(43,28),(43,41),(44,29),(44,35),(45,27),(45,52),(46,17),(46,31),(47,30),(47,37),(48,36),(48,42),(49,36),(49,37),(50,42),(50,51),(51,23),(51,57),(52,24),(52,46),(53,20),(53,56),(54,18),(54,53),(55,19),(55,54),(56,22),(56,52),(57,21),(57,55),(58,25),(58,51),(59,50),(59,58),(60,48),(60,49),(61,48),(61,50),(62,47),(62,49),(63,16),(63,58)],64)
=> ?
=> ?
=> ?
=> ? = 8
([(0,41),(1,40),(2,34),(3,45),(3,46),(4,45),(4,50),(5,44),(5,46),(6,34),(6,44),(7,40),(7,41),(7,50),(9,33),(10,31),(11,32),(12,16),(13,53),(14,54),(15,54),(16,8),(17,16),(18,22),(18,53),(19,23),(19,53),(20,24),(21,25),(22,29),(22,55),(23,30),(23,55),(24,26),(25,17),(26,39),(27,14),(27,52),(28,15),(28,52),(29,27),(29,51),(30,28),(30,51),(31,11),(31,42),(32,9),(32,36),(33,12),(33,17),(34,20),(35,38),(35,49),(36,25),(36,33),(37,24),(37,38),(38,26),(38,47),(39,14),(39,15),(40,13),(40,18),(41,13),(41,19),(42,32),(42,43),(43,21),(43,36),(44,20),(44,37),(45,35),(45,48),(46,35),(46,37),(47,27),(47,28),(47,39),(48,22),(48,23),(48,49),(49,29),(49,30),(49,47),(50,18),(50,19),(50,48),(51,42),(51,52),(52,43),(52,54),(53,10),(53,55),(54,21),(55,31),(55,51)],56)
=> ?
=> ?
=> ?
=> ? = 8
([(0,86),(1,74),(2,75),(3,93),(3,98),(4,92),(4,93),(5,74),(5,97),(6,75),(6,92),(7,86),(7,97),(7,98),(8,12),(10,11),(11,9),(12,10),(13,73),(14,91),(15,24),(15,90),(16,71),(17,70),(18,40),(19,21),(19,72),(20,23),(20,96),(21,46),(22,52),(23,84),(24,22),(24,79),(25,57),(25,66),(26,36),(26,65),(27,39),(27,68),(28,38),(28,67),(29,118),(30,100),(31,104),(32,105),(33,111),(34,113),(35,20),(35,115),(36,15),(36,101),(37,16),(37,111),(38,17),(38,112),(39,14),(39,110),(40,8),(41,55),(41,107),(42,58),(42,102),(43,34),(43,108),(44,72),(45,38),(45,106),(46,40),(47,53),(48,36),(48,104),(49,50),(50,44),(51,41),(51,100),(52,49),(53,76),(54,30),(54,102),(55,32),(55,99),(56,26),(56,48),(56,116),(57,28),(57,45),(57,114),(58,43),(58,109),(59,32),(59,113),(60,56),(60,119),(61,88),(62,45),(62,103),(63,29),(63,115),(64,31),(64,116),(65,37),(65,101),(66,35),(66,114),(67,60),(67,112),(68,25),(68,77),(68,110),(69,13),(69,83),(70,61),(71,69),(72,18),(72,46),(73,19),(73,44),(74,85),(75,47),(76,34),(76,59),(77,57),(77,62),(77,117),(78,55),(78,59),(78,108),(79,52),(79,82),(80,51),(80,87),(80,102),(81,53),(81,95),(82,49),(82,83),(83,50),(83,73),(84,31),(84,48),(85,30),(85,51),(86,42),(86,54),(87,41),(87,78),(87,109),(88,33),(88,37),(89,69),(89,82),(90,79),(90,89),(91,35),(91,63),(92,47),(92,81),(93,81),(93,94),(94,58),(94,87),(94,95),(95,43),(95,76),(95,78),(96,56),(96,64),(96,84),(97,54),(97,80),(97,85),(98,42),(98,80),(98,94),(99,105),(99,117),(100,39),(100,107),(101,90),(101,111),(102,27),(102,100),(102,109),(103,29),(103,106),(104,33),(104,101),(105,103),(106,112),(106,118),(107,99),(107,110),(108,77),(108,99),(108,113),(109,68),(109,107),(109,108),(110,66),(110,91),(110,117),(111,71),(111,89),(112,70),(112,119),(113,62),(113,105),(114,67),(114,106),(114,115),(115,60),(115,96),(115,118),(116,65),(116,88),(116,104),(117,63),(117,103),(117,114),(118,64),(118,119),(119,61),(119,116)],120)
=> ?
=> ?
=> ?
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12],[13],[14]]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,5,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12],[13],[14]]
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14],[15]]
=> ? = 6
([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ?
=> ?
=> ? = 4
([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ?
=> ?
=> ? = 4
([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ?
=> ?
=> ?
=> ? = 6
([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ?
=> ?
=> ? = 5
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ?
=> ?
=> ? = 5
([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ?
=> ?
=> ?
=> ? = 6
([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ?
=> ?
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5
([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6
([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ?
=> ?
=> ? = 4
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ?
=> ?
=> ? = 6
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ?
=> ?
=> ?
=> ? = 8
([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ?
=> ?
=> ? = 6
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,16),(1,20),(2,8),(2,16),(2,17),(3,9),(3,10),(3,17),(3,20),(4,12),(4,13),(4,16),(4,17),(4,20),(5,11),(5,13),(5,16),(5,17),(5,20),(6,8),(6,10),(6,11),(6,12),(6,20),(8,19),(8,23),(9,22),(10,15),(10,19),(10,22),(11,14),(11,15),(11,19),(11,23),(12,14),(12,15),(12,19),(12,23),(13,14),(13,22),(13,23),(14,18),(14,21),(15,18),(15,21),(16,22),(16,23),(17,19),(17,22),(17,23),(18,7),(19,18),(19,21),(20,15),(20,22),(20,23),(21,7),(22,21),(23,18),(23,21)],24)
=> ?
=> ?
=> ?
=> ? = 9
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ?
=> ?
=> ?
=> ? = 8
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,12),(2,16),(2,17),(3,11),(3,16),(3,17),(4,9),(4,10),(4,16),(5,9),(5,12),(5,13),(5,17),(6,1),(6,10),(6,11),(6,13),(6,17),(8,18),(8,20),(9,14),(9,21),(10,14),(10,19),(10,21),(11,19),(11,21),(12,15),(12,21),(13,8),(13,14),(13,15),(13,19),(14,18),(14,20),(15,18),(15,20),(16,21),(17,15),(17,19),(17,21),(18,7),(19,18),(19,20),(20,7),(21,20)],22)
=> ?
=> ?
=> ?
=> ? = 8
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8
([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 8
([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ?
=> ?
=> ? = 5
([(0,2),(0,3),(0,4),(0,5),(1,6),(1,16),(2,8),(2,10),(2,12),(3,8),(3,9),(3,11),(4,9),(4,10),(4,13),(5,1),(5,11),(5,12),(5,13),(6,17),(8,19),(9,14),(9,19),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(12,19),(13,6),(13,14),(13,15),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7
([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ?
=> ?
=> ?
=> ? = 7
([(0,2),(0,3),(0,4),(0,5),(0,6),(1,20),(1,22),(2,13),(2,14),(2,17),(3,11),(3,12),(3,17),(4,8),(4,9),(4,11),(4,17),(5,1),(5,8),(5,10),(5,14),(5,17),(6,9),(6,10),(6,12),(6,13),(8,15),(8,20),(8,22),(9,15),(9,19),(9,22),(10,15),(10,16),(10,19),(10,20),(11,22),(12,19),(12,22),(13,16),(13,19),(14,16),(14,20),(14,22),(15,18),(15,21),(16,18),(16,21),(17,19),(17,20),(17,22),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,21)],23)
=> ?
=> ?
=> ?
=> ? = 9
([(0,2),(0,3),(0,4),(0,5),(1,17),(2,8),(2,9),(2,12),(3,7),(3,9),(3,11),(4,7),(4,8),(4,10),(5,1),(5,10),(5,11),(5,12),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(10,17),(11,15),(11,16),(11,17),(12,14),(12,15),(12,17),(13,19),(14,18),(14,19),(15,18),(15,19),(16,18),(16,19),(17,18),(18,6),(19,6)],20)
=> ?
=> ?
=> ?
=> ? = 7
Description
The last entry in the first row of a standard tableau.
The following 217 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St000546The number of global descents of a permutation. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000678The number of up steps after the last double rise of a Dyck path. St000809The reduced reflection length of the permutation. St001462The number of factors of a standard tableaux under concatenation. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000925The number of topologically connected components of a set partition. St001777The number of weak descents in an integer composition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000172The Grundy number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000006The dinv of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000015The number of peaks of a Dyck path. St001530The depth of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000741The Colin de Verdière graph invariant. St001152The number of pairs with even minimum in a perfect matching. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000184The size of the centralizer of any permutation of given cycle type. St000228The size of a partition. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000384The maximal part of the shifted composition of an integer partition. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000544The cop number of a graph. St000553The number of blocks of a graph. St000667The greatest common divisor of the parts of the partition. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000784The maximum of the length and the largest part of the integer partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001286The annihilation number of a graph. St001312Number of parabolic noncrossing partitions indexed by the composition. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000008The major index of the composition. St000021The number of descents of a permutation. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000354The number of recoils of a permutation. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001340The cardinality of a minimal non-edge isolating set of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001814The number of partitions interlacing the given partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001812The biclique partition number of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001834The number of non-isomorphic minors of a graph. St001706The number of closed sets in a graph. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000806The semiperimeter of the associated bargraph. St001674The number of vertices of the largest induced star graph in the graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001323The independence gap of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001651The Frankl number of a lattice. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice. St001792The arboricity of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000535The rank-width of a graph. St000537The cutwidth of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001638The book thickness of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph.
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