  We present a full analytical solution of the multiconfigurational
strongly-correlated mixed-valence problem corresponding to the $N$-Hubbard ring
filled with $N-1$ electrons, and infinite on-site repulsion. While the
eigenvalues and the eigenstates of the model are known already, analytical
determination of their degeneracy is presented here for the first time. The
full solution, including degeneracy count, is achieved for each spin
configuration by mapping the Hubbard model into a set of Huckel-annulene
problems for rings of variable size. The number and size of these effective
Huckel annulenes, both crucial to obtain Hubbard states and their degeneracy,
are determined by solving a well-known combinatorial enumeration problem, the
necklace problem for $N-1$ beads and two colors, within each subgroup of the
$C_{N-1}$ permutation group. Symmetry-adapted solution of the necklace
enumeration problem is finally achieved by means of the subduction of coset
representation technique [S. Fujita, Theor. Chim. Acta 76, 247 (1989)], which
provides a general and elegant strategy to solve the one-hole infinite-$U$
Hubbard problem, including degeneracy count, for any ring size. The proposed
group theoretical strategy to solve the infinite-$U$ Hubbard problem for $N-1$
electrons, is easily generalized to the case of arbitrary electron count $L$,
by analyzing the permutation group $C_L$ and all its subgroups.
