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] |
The object isSperner is a method function.