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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St001827
Values
([],1)
=> 0
([],2)
=> 1
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([(0,2),(0,3),(1,2),(1,3)],4)
=> 6
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 10
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 15
([],5)
=> 0
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 1
([(0,1),(2,4),(3,4)],5)
=> 1
([(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 10
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 9
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 18
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 20
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 32
([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 9
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 18
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 10
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 19
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 32
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 18
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 16
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 31
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 51
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 33
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 52
Description
The number of two-component spanning forests of a graph.
A '''spanning subgraph''' is a subgraph which contains all vertices of the ambient graph. A '''forest''' is a graph which contains no cycles, and has any number of connected components. A '''two-component spanning forest''' is a spanning subgraph which contains no cycles and has two connected components.
Matching statistic: St001876
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0
([],2)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1}
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1}
([],3)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,2,3}
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,1,2,3}
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,2,3}
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,3}
([],4)
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([],5)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,26),(17,26),(18,26),(19,26),(20,26),(21,26),(22,26),(23,26),(24,26),(25,26)],27)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
=> ? ∊ {0,0,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> ([],1)
=> ([],1)
=> ? ∊ {0,0,0,0,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000175
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000205
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000225
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000319
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000506
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 11%●distinct values known / distinct values provided: 2%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {0,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {0,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
The number of standard desarrangement tableaux of shape equal to the given partition.
A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
* [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
* [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St000749
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 11%●distinct values known / distinct values provided: 1%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 0
([],2)
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,1}
([],3)
=> [1,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,2,3}
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,2,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,3,3,3,6,6,10,15}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? ∊ {1,1,1,3,3,4,4,4,4,8,9,9,9,10,10,16,18,18,18,19,20,31,32,32,33,51,52,77,110}
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,4,4,5,5,5,5,5,5,5,8,8,8,9,9,11,12,12,12,12,12,12,12,12,14,14,14,14,15,15,16,16,20,21,24,26,26,26,26,26,26,26,27,27,27,30,30,30,30,31,32,32,33,34,40,45,47,47,47,52,52,53,53,53,54,54,57,57,57,58,58,59,59,59,62,63,64,65,75,91,91,91,93,96,97,97,98,98,99,99,99,103,104,105,105,108,109,111,117,125,152,152,160,161,161,162,168,168,169,170,174,176,176,180,235,254,255,256,262,264,270,271,272,378,387,389,398,408,561,572,792,1080}
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [1]
=> 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1]
=> 0
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition.
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