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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001369
St001369: Finite Cartan types ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> 1 = 0 + 1
['A',2]
=> 1 = 0 + 1
['B',2]
=> 2 = 1 + 1
['G',2]
=> 3 = 2 + 1
['A',3]
=> 1 = 0 + 1
['B',3]
=> 2 = 1 + 1
['C',3]
=> 2 = 1 + 1
['A',4]
=> 1 = 0 + 1
['D',4]
=> 2 = 1 + 1
['A',5]
=> 1 = 0 + 1
['A',6]
=> 1 = 0 + 1
Description
The largest coefficient in the highest root in the root system of a Cartan type.
Matching statistic: St001031
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001031: 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]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
['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]
=> 0
['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]
=> 1
['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]
=> 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)
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [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]
=> 0
['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]
=> 0
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St000150
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> ([],1)
=> [1]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 0
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> [7,1,1]
=> 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)
=> ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> [9,1]
=> 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)
=> ([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> 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)
=> ([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> 0
['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)
=> ([(1,2),(1,7),(1,16),(1,18),(1,20),(2,6),(2,15),(2,17),(2,19),(3,6),(3,7),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(4,5),(4,11),(4,12),(4,14),(4,15),(4,17),(4,18),(4,19),(4,20),(5,11),(5,12),(5,13),(5,16),(5,17),(5,18),(5,19),(5,20),(6,7),(6,9),(6,11),(6,13),(6,16),(6,18),(6,20),(7,10),(7,12),(7,14),(7,15),(7,17),(7,19),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(9,10),(9,12),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(10,11),(10,13),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(11,12),(11,14),(11,15),(11,17),(11,19),(11,20),(12,13),(12,16),(12,18),(12,19),(12,20),(13,14),(13,15),(13,17),(13,18),(13,19),(13,20),(14,16),(14,17),(14,18),(14,19),(14,20),(15,16),(15,18),(15,20),(16,17),(16,19),(17,18),(17,20),(18,19),(19,20)],21)
=> [20,1]
=> ? = 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St001314
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
['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]
=> 0
['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]
=> ? = 1
['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]
=> ? = 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)
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [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]
=> ? = 0
['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]
=> ? = 0
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
Matching statistic: St001646
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> 2
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,6),(3,7),(3,8),(4,5),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,6),(3,7),(3,8),(4,5),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 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)
=> ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 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)
=> ([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 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)
=> ([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 0
['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)
=> ([(1,2),(1,7),(1,16),(1,18),(1,20),(2,6),(2,15),(2,17),(2,19),(3,6),(3,7),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(4,5),(4,11),(4,12),(4,14),(4,15),(4,17),(4,18),(4,19),(4,20),(5,11),(5,12),(5,13),(5,16),(5,17),(5,18),(5,19),(5,20),(6,7),(6,9),(6,11),(6,13),(6,16),(6,18),(6,20),(7,10),(7,12),(7,14),(7,15),(7,17),(7,19),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(9,10),(9,12),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(10,11),(10,13),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(11,12),(11,14),(11,15),(11,17),(11,19),(11,20),(12,13),(12,16),(12,18),(12,19),(12,20),(13,14),(13,15),(13,17),(13,18),(13,19),(13,20),(14,16),(14,17),(14,18),(14,19),(14,20),(15,16),(15,18),(15,20),(16,17),(16,19),(17,18),(17,20),(18,19),(19,20)],21)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(2,17),(2,18),(2,19),(2,20),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(4,20),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(7,20),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(15,16),(15,17),(15,18),(15,19),(15,20),(16,17),(16,18),(16,19),(16,20),(17,18),(17,19),(17,20),(18,19),(18,20),(19,20)],21)
=> ? = 0
Description
The number of edges that can be added without increasing the maximal degree of a graph.
This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on $n$ vertices, which has maximal degree $n-1$ and therefore has statistic $\binom{n-1}{2}$.
Matching statistic: St000785
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> 3 = 2 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,6),(3,7),(3,8),(4,5),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> ([(2,7),(3,5),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,6),(3,7),(3,8),(4,5),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1 + 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)
=> ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 0 + 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)
=> ([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 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)
=> ([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 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)
=> ([(1,2),(1,7),(1,16),(1,18),(1,20),(2,6),(2,15),(2,17),(2,19),(3,6),(3,7),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(4,5),(4,11),(4,12),(4,14),(4,15),(4,17),(4,18),(4,19),(4,20),(5,11),(5,12),(5,13),(5,16),(5,17),(5,18),(5,19),(5,20),(6,7),(6,9),(6,11),(6,13),(6,16),(6,18),(6,20),(7,10),(7,12),(7,14),(7,15),(7,17),(7,19),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(9,10),(9,12),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(10,11),(10,13),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(11,12),(11,14),(11,15),(11,17),(11,19),(11,20),(12,13),(12,16),(12,18),(12,19),(12,20),(13,14),(13,15),(13,17),(13,18),(13,19),(13,20),(14,16),(14,17),(14,18),(14,19),(14,20),(15,16),(15,18),(15,20),(16,17),(16,19),(17,18),(17,20),(18,19),(19,20)],21)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(2,17),(2,18),(2,19),(2,20),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(4,20),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(7,20),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(15,16),(15,17),(15,18),(15,19),(15,20),(16,17),(16,18),(16,19),(16,20),(17,18),(17,19),(17,20),(18,19),(18,20),(19,20)],21)
=> ? = 0 + 1
Description
The number of distinct colouring schemes of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the number of distinct partitions that occur.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$.
Matching statistic: St001503
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001503: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001503: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [1]
=> [1,0]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 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]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
['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]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1 + 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)
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 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)
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 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)
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 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]
=> [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
Description
The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
Matching statistic: St001739
Values
['A',1]
=> ([],1)
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 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)
=> 2 = 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)
=> 3 = 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)
=> 1 = 0 + 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)
=> ? = 1 + 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)
=> ? = 1 + 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)
=> ? = 0 + 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)
=> ? = 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)
=> ([(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)
=> ? = 0 + 1
Description
The number of graphs with the same edge polytope as the given graph.
The symmetric edge polytope of a graph on $n$ vertices is the polytope in $\mathbb R^n$ defined as the convex hull of $e_i + e_j$ for each edge $(i, j)$, where $e_1,\dots, e_n$ denotes the standard basis.
Matching statistic: St001526
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
['A',1]
=> ([],1)
=> [1]
=> [1,0,1,0]
=> 2 = 0 + 2
['A',2]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
['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 + 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]
=> 2 = 0 + 2
['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]
=> ? = 1 + 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]
=> ? = 1 + 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]
=> 2 = 0 + 2
['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 + 2
['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]
=> ? = 0 + 2
['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]
=> ? = 0 + 2
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001442
Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001442: Integer partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
['A',1]
=> ([],1)
=> [2]
=> 1 = 0 + 1
['A',2]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 0 + 1
['B',2]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2 = 1 + 1
['G',2]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 3 = 2 + 1
['A',3]
=> ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> ? = 0 + 1
['B',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> ? = 1 + 1
['C',3]
=> ([(0,7),(1,8),(2,7),(2,8),(4,5),(5,3),(6,5),(7,6),(8,4),(8,6)],9)
=> [6,6,6,2]
=> ? = 1 + 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)
=> [10,10,10,5,5,2]
=> ? = 0 + 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)
=> [6,6,6,6,6,6,6,3,3,2]
=> ? = 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)
=> [12,12,12,12,12,12,12,12,12,6,6,6,4,2]
=> ? = 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)
=> [14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,7,7,7,7,7,2]
=> ? = 0 + 1
Description
The number of standard Young tableaux whose major index is divisible by the size of a given integer partition.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001964The interval resolution global dimension of a poset. St001578The minimal number of edges to add or remove to make a graph a line graph. St001624The breadth of a lattice. St001783The number of odd automorphisms of a graph.
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