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

## Synopsis

• Usage:
i = isSperner P
• Inputs:
• P, an instance of the type Poset, a ranked poset
• Outputs:
• i, , whether $P$ is Sperner

## 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]