  The existence of the Nagaoka ferromagnetism is examined in the context of the
one-dimensional $U=\infty$ Hubbard model. We construct the exact quantum
partition function to describe the physics of such a regime. Our calculation
reveals that, while the ground state in an open chain is always
spin-degenerate, in a finite size closed chain with at least one vacancy, the
ground state can only be ferromagnetic when the number of electrons is less or
equal to three. Our results shed more light on a very recent experimental
verification of Nagaoka ferromagnetism in a quantum dot set up.
