Schläfli symbol

The Schläfli symbol mainly is a notation for regular polytopes. For example the dodecahedron is {5,3}. That one just describes a regular polyhedron with pentagonal faces, three per vertex. It well is applicable to higher dimensions as well. E.g. a regular polychoron having R cells of type {P, Q} around each of its ridges (here: edges) will be denoted {P, Q, R}.

Conversion to Coxeter-Dynkin diagrams
Even so it is a historically older notation than that of the Coxeter-Dynkin diagrams it is closely related. In fact it was its precursor. Whenever xPoQo...oRo is the latter then {P, Q, ..., R} would be the former, or conversely. In fact a single ringed node (x symbol in the typewriter friendly version) always is being assumed to be at the very left.

Coxeter adapted the Schläfli symbol lateron to quasiregular polytopes as well: Again a single ringed node would occur within its Coxeter-Dynkin diagram. Then the diagram gets folded and displayed such that its single ringed node is displayed to the very left, while the two legs would run to the right. The according Schläfli symbol then simply is a left curly bracket followed by the two left-aligned number sequences of those legs each, one atop the other, and finally closed by the right curly bracket. He further even allowed thereby for bifurcation nodes in the diagram (as being used for the Gosset polytopes). Then any of the left-aligned number sequences might split up into two (or more) lines from some point on.

Sure, none of these extended usages is better usable within inline texts as the according graphical devices of the Coxeter-Dynkin diagrams. Moreover they always are restricted to a single ringed node, while the latter are much more general. This is why the Schläfli symbols mostly became abandoned in favor of the latter.