I'm trying to refresh things I learned ages ago, and I believe I either never learned the proper mathematical background for multi-particle states or I forgot the details. I intuitively write things like $\newcommand{\ket}[1]{\left|#1\right\rangle}\ket{\uparrow\downarrow}$ and $\ket{n_1,n_2}$ and remember the latter being related to multi-mode Fock states, but sometimes also tensor products $\ket 1\otimes\ket 4$ and maybe even (vectors of Hilbert) vectors $\left(\ket a\atop \ket b\right)$ (or $\ket{\left(a\atop b\right)}$?). But are all those different but equivalent notations for the same mathematical concepts and merely a matter of taste/convenience/legibility, or are there severe differences and something like $\left(\ket{a,2}\atop \ket b \otimes \ket {42}\right)$ actually has a sensible meaning?

While I should sooner or later (re)learn the details including representation theory in (or "of"?) Hilbert spaces etc, for now I'd appreciate pointers into the proper directions.