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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 :

For the programmer

The object isSperner is a method function.