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We study the action of $S_n$ on the set of break divisors on completemultigraphs $K_{n}^m$. We provide an alternative characterization for thesedivisors, by virtue of which we show that orbits of this action are enumeratedby the numerical Donaldson-Thomas invariants of $(m+1)$-loop quivers. Ourcharacterization also allows us to restrict this action to $S_{n-1}$ and weidentify the resulting $S_{n-1}$-module as that afforded by $K_n^m$-parkingfunctions.

Divisors on complete multigraphs and Donaldson-Thomas invariants of loop quivers