Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000852
Mapping
St000852: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => 3
['A',2]
 => 12
['B',2]
 => 15
['G',2]
 => 21
['A',3]
 => 55
['B',3]
 => 84
['C',3]
 => 84
['A',4]
 => 273
['B',4]
 => 495
['C',4]
 => 495
['D',4]
 => 336
['F',4]
 => 780
['A',5]
 => 1428
['B',5]
 => 3003
['C',5]
 => 3003
['D',5]
 => 2079
['A',6]
 => 7752
['B',6]
 => 18564
['C',6]
 => 18564
['D',6]
 => 13013
['E',6]
 => 16588
['A',7]
 => 43263
['B',7]
 => 116280
['C',7]
 => 116280
['D',7]
 => 82212
['E',7]
 => 144210
['A',8]
 => 246675
['B',8]
 => 735471
['C',8]
 => 735471
['D',8]
 => 523260
['E',8]
 => 1520922
Description
The second Fuss-Catalan number of a finite Cartan type. The Fuss-Catalan numbers of a finite Cartan type are given by $$\frac{1}{|W|}\prod (d_i+mh) = \prod \frac{d_i+mh}{d_i}$$ where the products run over all degrees of homoneneous fundamenal invariants of the Weyl group of a Cartan type.
Matching statistic: St000063
(load all 2 compositions to match this statistic)
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 16%
Values
['A',1]
 => ([],1)
 => [2]
 => 3
['A',2]
 => ([(0,2),(1,2)],3)
 => [3,2]
 => 12
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 15
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => 21
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [8,4,2]
 => ? = 55
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [6,6,6,2]
 => ? = 84
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [6,6,6,2]
 => ? = 84
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [10,10,10,5,5,2]
 => ? = 273
['B',4]
 => ([(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)
 => [8,8,8,8,8,8,8,8,4,2]
 => ? = 495
['C',4]
 => ([(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)
 => [8,8,8,8,8,8,8,8,4,2]
 => ? = 495
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [6,6,6,6,6,6,6,3,3,2]
 => ? = 336
['F',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)
 => [12,12,12,12,12,12,12,12,4,3,2]
 => ? = 780
['A',5]
 => ([(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)
 => [12,12,12,12,12,12,12,12,12,6,6,6,4,2]
 => ? = 1428
['B',5]
 => ([(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)
 => [10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,2]
 => ? = 3003
['C',5]
 => ([(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)
 => [10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,2]
 => ? = 3003
['D',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)
 => [16,16,16,16,16,16,16,8,8,8,8,8,8,8,8,4,2]
 => ? = 2079
['A',6]
 => ([(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)
 => [14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,7,7,7,7,7,2]
 => ? = 7752
['B',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)
 => ?
 => ? = 18564
['C',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)
 => ?
 => ? = 18564
['D',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)
 => ?
 => ? = 13013
['E',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)
 => ?
 => ? = 16588
['A',7]
 => ([(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)
 => ?
 => ? = 43263
['B',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)
 => ?
 => ? = 116280
['C',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)
 => ?
 => ? = 116280
['D',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)
 => ?
 => ? = 82212
['E',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)
 => ?
 => ? = 144210
['A',8]
 => ([(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)
 => ?
 => ? = 246675
['B',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)
 => ?
 => ? = 735471
['C',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)
 => ?
 => ? = 735471
['D',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)
 => ?
 => ? = 523260
['E',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)
 => ?
 => ? = 1520922
Description
The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.