  The antiferromagnetic Heisenberg spin chain of odd spin $S$ is in the Haldane
phase with several defining physical properties, such as thermodynamical
ground-state degeneracy, symmetry-protected edge states, and nonzero string
order parameter. If nonzero hole concentration $\delta$ and hole hopping energy
$t$ are considered, the spin chain is replaced by a spin-$S$ $t$-$J$ chain. The
motivation of this paper is to generalize the discussions of the Haldane phase
to the doped spin chain. The \emph{first result} of this paper is that, for the
model considered here, the $\mathbb{Z}_2$ sign structure in the usual Ising
basis can be totally removed by two consecutive unitary transformations
consisting of a spatially local one and a nonlocal one. Direct from the sign
structure, the \emph{second result} of this paper is that the Marshall theorem
and the Lieb-Mattis theorem for pure spin systems are generalized to the
$t$-$J$ chain for arbitrary $S$ and $\delta$. A corollary of the theorem
provides us with the ground-state degeneracy in the thermodynamic limit. The
\emph{third result} of this paper is about the phase diagram. We show that the
defining properties of the Haldane phase survive in the small $t/J$ limit. The
large $t/J$ phase supports a gapped spin sector with similar properties
(ground-state degeneracy, edge state, and string order parameter) of the
Haldane chain, although the charge sector is gapless.
