Your data matches 139 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000142
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => 0
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => 0
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => 1
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => 3
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => 0
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => 0
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => 0
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => 2
Description
The number of even parts of a partition.
Matching statistic: St001092
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => 0
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => 0
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => 1
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => 3
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => 0
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => 0
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => 0
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => 2
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St001586
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ? ∊ {0,0,0,2,3}
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001587
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ? ∊ {0,0,0,2,3}
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St000150
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,1,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [3,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [5,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,1,1,1]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [5,3,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [5,3,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [3,2,2,1,1,1]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => [7,5,3,1]
 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => [7,5,3,1]
 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => [6,5,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ?
 => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => [5,3,2,2,1,1,1]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ?
 => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ?
 => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ?
 => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ?
 => ? ∊ {0,0,0,2,3}
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000257
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,1,1]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [3,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [5,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,1,1,1]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [5,3,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [5,3,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [3,2,2,1,1,1]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => [7,5,3,1]
 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => [7,5,3,1]
 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => [6,5,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ?
 => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => [5,3,2,2,1,1,1]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ?
 => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ?
 => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ?
 => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ?
 => ? ∊ {0,0,0,2,3}
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000292
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => 1 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 01 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 11 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 11 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 0101 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 1111 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 1111 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 1111 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ? => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 10101 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ? => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ? => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ? => ? ∊ {0,0,0,2,3}
Description
The number of ascents of a binary word.
Matching statistic: St001355
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => 1 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 01 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 11 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 11 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 0101 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 1111 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 1111 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 1111 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ? => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 10101 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ? => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ? => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ? => ? ∊ {0,0,0,2,3}
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Matching statistic: St001420
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001420: Binary words ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => 1 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => 01 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => 11 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => 11 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => 101 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => 111 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => 0101 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 1111 => 0
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => 1111 => 0
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => 1111 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ? => ? = 0
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => 10101 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ? => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ? => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ? => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ? => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ? => ? ∊ {0,0,0,2,3}
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Matching statistic: St000149
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 75%
Values
['A',1]
 => ([],1)
 => [1]
 => [1]
 => 0
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [3]
 => 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [2,1,1]
 => 0
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [2,2,1,1]
 => 0
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,3]
 => 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 0
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => [5,3,1]
 => [3,2,2,1,1]
 => 0
['A',4]
 => ([(0,8),(1,7),(2,7),(2,9),(3,8),(3,9),(5,4),(6,4),(7,5),(8,6),(9,5),(9,6)],10)
 => [4,3,2,1]
 => [5,5]
 => 2
['B',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? ∊ {0,0,0}
['C',4]
 => ([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
 => [7,5,3,1]
 => [4,3,3,2,2,1,1]
 => ? ∊ {0,0,0}
['D',4]
 => ([(0,10),(1,9),(2,8),(3,8),(3,9),(3,10),(5,11),(6,11),(7,11),(8,5),(8,6),(9,5),(9,7),(10,6),(10,7),(11,4)],12)
 => [5,3,3,1]
 => [3,3,2,2,1,1]
 => 0
['F',4]
 => ([(0,18),(1,19),(2,18),(2,22),(3,19),(3,22),(4,6),(6,5),(7,11),(8,16),(9,17),(10,13),(10,14),(11,4),(12,23),(13,8),(13,23),(14,9),(14,23),(15,11),(16,15),(17,7),(17,15),(18,20),(19,21),(20,12),(20,13),(21,12),(21,14),(22,10),(22,20),(22,21),(23,16),(23,17)],24)
 => [11,7,5,1]
 => ?
 => ? ∊ {0,0,0}
['A',5]
 => ([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
 => [5,4,3,2,1]
 => [5,5,5]
 => 2
['B',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ?
 => ? ∊ {0,0,1}
['C',5]
 => ([(0,18),(1,17),(2,18),(2,24),(3,23),(3,24),(4,17),(4,23),(6,15),(7,16),(8,9),(9,5),(10,12),(11,13),(12,11),(13,14),(14,9),(15,7),(15,21),(16,8),(16,14),(17,10),(18,6),(18,19),(19,15),(19,22),(20,12),(20,22),(21,13),(21,16),(22,11),(22,21),(23,10),(23,20),(24,19),(24,20)],25)
 => [9,7,5,3,1]
 => ?
 => ? ∊ {0,0,1}
['D',5]
 => ([(0,13),(1,16),(2,15),(3,13),(3,17),(4,15),(4,16),(4,17),(6,10),(7,19),(8,19),(9,18),(10,5),(11,7),(11,18),(12,8),(12,18),(13,14),(14,7),(14,8),(15,9),(15,11),(16,9),(16,12),(17,11),(17,12),(17,14),(18,6),(18,19),(19,10)],20)
 => [7,5,4,3,1]
 => ?
 => ? ∊ {0,0,1}
['A',6]
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,13),(4,18),(5,14),(5,19),(7,9),(8,10),(9,11),(10,12),(11,6),(12,6),(13,7),(14,8),(15,9),(15,17),(16,10),(16,17),(17,11),(17,12),(18,7),(18,15),(19,8),(19,16),(20,15),(20,16)],21)
 => [6,5,4,3,2,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['B',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['C',6]
 => ([(0,23),(1,22),(2,23),(2,34),(3,33),(3,34),(4,33),(4,35),(5,22),(5,35),(7,20),(8,19),(9,21),(10,11),(11,6),(12,16),(13,15),(14,13),(15,17),(16,14),(17,18),(18,11),(19,9),(19,27),(20,8),(20,28),(21,10),(21,18),(22,12),(23,7),(23,24),(24,20),(24,32),(25,16),(25,31),(26,31),(26,32),(27,17),(27,21),(28,19),(28,29),(29,15),(29,27),(30,13),(30,29),(31,14),(31,30),(32,28),(32,30),(33,25),(33,26),(34,24),(34,26),(35,12),(35,25)],36)
 => [11,9,7,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['D',6]
 => ([(0,24),(1,23),(2,20),(3,22),(3,26),(4,20),(4,22),(5,23),(5,24),(5,26),(7,12),(8,19),(9,27),(10,29),(11,29),(12,6),(13,16),(13,27),(14,17),(14,27),(15,21),(16,10),(16,28),(17,11),(17,28),(18,12),(19,7),(19,18),(20,15),(21,10),(21,11),(22,15),(22,25),(23,9),(23,13),(24,9),(24,14),(25,16),(25,17),(25,21),(26,13),(26,14),(26,25),(27,8),(27,28),(28,19),(28,29),(29,18)],30)
 => [9,7,5,5,3,1]
 => ?
 => ? ∊ {0,0,0,2,3}
['E',6]
 => ([(0,28),(1,24),(2,23),(3,23),(3,29),(4,24),(4,30),(5,28),(5,29),(5,30),(6,7),(8,19),(9,20),(10,14),(10,15),(11,34),(12,32),(13,33),(14,8),(14,35),(15,9),(15,35),(16,6),(17,12),(17,31),(18,13),(18,31),(19,16),(20,16),(21,11),(21,32),(22,11),(22,33),(23,26),(24,27),(25,21),(25,22),(25,31),(26,12),(26,21),(27,13),(27,22),(28,17),(28,18),(29,17),(29,25),(29,26),(30,18),(30,25),(30,27),(31,10),(31,32),(31,33),(32,14),(32,34),(33,15),(33,34),(34,35),(35,19),(35,20)],36)
 => [11,8,7,5,4,1]
 => ?
 => ? ∊ {0,0,0,2,3}
Description
The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].
The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001423The number of distinct cubes in a binary word. St000143The largest repeated part of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001568The smallest positive integer that does not appear twice in the partition. St000992The alternating sum of the parts of an integer partition. St000137The Grundy value of an integer partition. St000148The number of odd parts of a partition. St000671The maximin edge-connectivity for choosing a subgraph. St001383The BG-rank of an integer partition. St001969The difference in the number of possibilities of choosing a pair of negative eigenvalues and the signature of a graph. St000273The domination number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001829The common independence number of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000091The descent variation of a composition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001271The competition number of a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001479The number of bridges of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001691The number of kings in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001797The number of overfull subgraphs of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001871The number of triconnected components of a graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000917The open packing number of a graph. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001111The weak 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001256Number of simple reflexive modules that are 2-stable reflexive. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001393The induced matching number of a graph. St001642The Prague dimension of a graph. St001716The 1-improper chromatic number of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000469The distinguishing number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001060The distinguishing index of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001570The minimal number of edges to add to make a graph Hamiltonian. St001638The book thickness of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001118The acyclic chromatic index of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001820The size of the image of the pop stack sorting operator. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000181The number of connected components of the Hasse diagram for the poset. St000455The second largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St001624The breadth of a lattice. St001783The number of odd automorphisms of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000286The number of connected components of the complement of a graph. St000311The number of vertices of odd degree in a graph.