I don’t know enough about defining Iterators to solve this 😞

I believe that the bug is

function Base.iterate(p::IntegerPartitions, xs = Int[])
    length(xs) == p.n && return
    xs = nextpartition(p.n,xs)
    (xs, xs)
end

since length(xs)==0 this returns nothing. I don't know how to have iterate here return [[]] only once.

This does the trick, needs some test etc.

function Base.iterate(p::IntegerPartitions, xs = Int[])
    length(xs) >= p.n + (p.n == 0) && return nothing
    xs = nextpartition(p.n,xs)
    (xs, xs)
end

Ah, no, this returns [0] which is not wrong but not "normalized"

Then maybe add this as a second function:

function Base.iterate(p::IntegerPartitions)
   p.n == 0 && return (Int[], Int[])
  return iterate(p, Int[])
end

and change the signature of the other function from function Base.iterate(p::IntegerPartitions, xs = Int[]) to function Base.iterate(p::IntegerPartitions, xs).