Some asymptotic properties of the set of values obtained by the central characters that correspond to single-cycle conjugacy classes of the symmetric group $S_n$ have been explored. Conjectures concerning the asymptotic behavior of the number of distinct central characters corresponding to the class of transpositions and that of three-cycles, are presented. Refinements involving partitioning the set of irreducible representations according to their Durfee box sizes are also examined. The technique used is brute-force enumeration; the conjectures presented suggest interesting asymptotic features of the representation theory of the symmetric group, yet to be uncovered.