The Greene-Kleitman partition $l$ of $P$ is the partition such that the sum of the first $k$ parts of $l$ is the maximum number of elements in a union of $k$ chains in $P$.
i1 : P = poset {{1,2},{2,3},{3,4},{2,5},{6,3}};
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i2 : greeneKleitmanPartition P
o2 = Partition{4, 2}
o2 : Partition
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The conjugate of $l$ has the same property, but with chains replaced by antichains. Because of this, it is often better to count via antichains instead of chains. This can be done by passing "antichains" as the Strategy.
i3 : D = dominanceLattice 6; |
i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
-- used 0.558115 seconds
o4 = Partition{9, 2}
o4 : Partition
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i5 : time greeneKleitmanPartition(D, Strategy => "chains")
-- used 6.718e-6 seconds
o5 = Partition{9, 2}
o5 : Partition
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The Greene-Kleitman partition of the $n$ chain is the partition of $n$ with $1$ part.
i6 : greeneKleitmanPartition chain 10
o6 = Partition{10}
o6 : Partition
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The object greeneKleitmanPartition is a method function with options.