Processing math: 35%

Identifier
Values
['A',1] => ([],1) => ([],1) => [1] => 1
['A',2] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => [3] => 1
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => [4] => 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) => [6] => 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) => [6] => 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) => [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) => ([(0,8),(1,7),(2,6),(3,6),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8)],9) => [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) => ([(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) => [10] => 1
['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) => ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16) => [16] => 1
['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) => ([(0,15),(1,11),(2,10),(3,13),(3,15),(4,14),(4,15),(5,10),(5,13),(6,11),(6,14),(7,8),(7,9),(7,12),(8,10),(8,13),(9,11),(9,14),(12,13),(12,14),(12,15)],16) => [16] => 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) => [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) => ([(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) => [15] => 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) => ([(0,17),(1,16),(2,11),(3,10),(4,10),(4,18),(5,11),(5,19),(6,16),(6,17),(6,18),(7,16),(7,17),(7,19),(8,12),(8,13),(8,14),(9,12),(9,13),(9,15),(10,12),(11,13),(12,18),(13,19),(14,16),(14,18),(14,19),(15,17),(15,18),(15,19)],20) => [20] => 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) => [21] => 1
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Description
The Kreweras number of an integer partition.
This is defined for λn with k parts as
\frac{1}{n+1}\binom{n+1}{n+1-k,\mu_1(\lambda),\ldots,\mu_n(\lambda)}
where \mu_j(\lambda) denotes the number of parts of \lambda equal to j, see [1]. This formula indeed counts the number of noncrossing set partitions where the ordered block sizes are the partition \lambda.
These numbers refine the Narayana numbers N(n,k) = \frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1} and thus sum up to the Catalan numbers \frac{1}{n+1}\binom{2n}{n}.
Map
to root poset
Description
The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where \alpha \prec \beta if \beta - \alpha is a simple root.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
Map
to graph
Description
Returns the Hasse diagram of the poset as an undirected graph.