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Posets :: isSperner

isSperner -- determines if a ranked poset has the Sperner property

Synopsis

Description

The ranked poset P is Sperner if the maximum size of a set of elements with the same rank is the dilworthNumber of P. That is, P is Sperner if the maximum size of a set of elements with the same rank is the maximum size of an antichain.

The n chain and the n booleanLattice are Sperner.

i1 : n = 5;
i2 : isSperner chain n

o2 = true
i3 : isSperner booleanLattice n

o3 = true

However, the following poset is non-Sperner as it has an antichain of size 4 but the set of elements of rank 0 and the set of elements of rank 1 are both of size 3.

i4 : P = poset {{1,4}, {1,5}, {1,6}, {2,6}, {3,6}};
i5 : isSperner P

o5 = false
i6 : isAntichain(P, {2,3,4,5})

o6 = true
i7 : rankGeneratingFunction P

o7 = 3q + 3

o7 : ZZ[q]

See also

Ways to use isSperner :