Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001038
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
 => ([],1)
 => [1]
 => [1,0,1,0]
 => 2
['A',2]
 => ([(0,2),(1,2)],3)
 => [2,1]
 => [1,0,1,0,1,0]
 => 3
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 4
['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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2
['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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 2
['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]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
['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]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 1
['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]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
['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]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000274
Mapping
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000274: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 43%
Values
['A',1]
 => ([],1)
 => ([],1)
 => ([(0,1)],2)
 => 1 = 2 - 1
['A',2]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
['B',2]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
['G',2]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
['A',3]
 => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 4 - 1
['B',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 - 1
['C',3]
 => ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
 => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
 => ([(0,8),(0,9),(1,7),(1,9),(2,6),(2,9),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2 - 1
['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)
 => ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
 => ([(0,8),(0,10),(1,7),(1,10),(2,5),(2,6),(2,10),(3,7),(3,9),(3,10),(4,8),(4,9),(4,10),(5,7),(5,9),(5,10),(6,8),(6,9),(6,10),(7,10),(8,10),(9,10)],11)
 => ? = 5 - 1
['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)
 => ([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)
 => ([(0,11),(0,12),(1,10),(1,12),(2,9),(2,12),(3,8),(3,12),(4,8),(4,9),(4,10),(4,12),(5,8),(5,9),(5,11),(5,12),(6,8),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 - 1
['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)
 => ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
 => ([(0,11),(0,15),(1,10),(1,15),(2,8),(2,9),(2,15),(3,10),(3,13),(3,15),(4,11),(4,14),(4,15),(5,13),(5,14),(5,15),(6,8),(6,10),(6,13),(6,15),(7,9),(7,11),(7,14),(7,15),(8,12),(8,15),(9,12),(9,15),(10,15),(11,15),(12,13),(12,14),(12,15),(13,15),(14,15)],16)
 => ? = 6 - 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)
 => ([(0,14),(1,13),(2,18),(2,20),(3,19),(3,20),(4,11),(4,12),(5,13),(5,18),(6,14),(6,19),(7,9),(7,13),(7,18),(8,10),(8,14),(8,19),(9,11),(9,15),(10,12),(10,16),(11,17),(12,17),(15,17),(15,18),(15,20),(16,17),(16,19),(16,20)],21)
 => ([(0,14),(0,21),(1,13),(1,21),(2,18),(2,20),(2,21),(3,19),(3,20),(3,21),(4,11),(4,12),(4,21),(5,13),(5,18),(5,21),(6,14),(6,19),(6,21),(7,9),(7,13),(7,18),(7,21),(8,10),(8,14),(8,19),(8,21),(9,11),(9,15),(9,21),(10,12),(10,16),(10,21),(11,17),(11,21),(12,17),(12,21),(13,21),(14,21),(15,17),(15,18),(15,20),(15,21),(16,17),(16,19),(16,20),(16,21),(17,21),(18,21),(19,21),(20,21)],22)
 => ? = 7 - 1
Description
The number of perfect matchings of a graph. A matching of a graph $G$ is a subset $F \subset E(G)$ such that no two edges in $F$ share a vertex in common. A perfect matching $F'$ is then a matching such that every vertex in $V(G)$ is incident with exactly one edge in $F'$.