  We prove an identity between three infinite families of polynomials which are
defined in terms of `bosonic', `fermionic', and `one-dimensional configuration'
sums. In the limit where the polynomials become infinite series, they give
different-looking expressions for the characters of the two integrable
representations of the affine $su(2)$ algebra at level one. We conjecture yet
another fermionic sum representation for the polynomials which is constructed
directly from the Bethe-Ansatz solution of the Heisenberg spin chain.
