On General Principal Symmetric Ideals

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is sufficiently large. In this paper, we strengthen that result by giving a effective bound on the number of variables needed for their conclusion to hold. The bound is related to a well-known integer sequence involving partition numbers (OEIS A000070). Along the way, we prove a recognition theorem for principal symmetric ideals. We also introduce the class of maximal $r$-generated submodules, determine their structure, and connect them to general symmetric ideals.