  There is no guarantee that the violation of lepton number, assuming it
exists, will primarily manifest itself in neutrinoless double beta decay
($0\nu\beta\beta$). Lepton-number violation and lepton-flavor violation may be
related, and much-needed information regarding the physics that violates lepton
number can be learned by exploring observables that violate lepton flavors
other than electron flavor. One of the most promising observables is $\mu^-
\rightarrow e^+$ conversion, which can be explored by experiments that are
specifically designed to search for $\mu^- \rightarrow e^-$ conversion. We
survey lepton-number-violating dimension-five, -seven, and -nine effective
operators in the standard model and discuss the relationships between Majorana
neutrino masses and the rates for $0\nu\beta\beta$ and $\mu^- \rightarrow e^+$
conversion. While $0\nu\beta\beta$ has the greatest sensitivity to new
ultraviolet energy scales, its rate might be suppressed by the new physics
relationship to lepton flavor, and $\mu^- \rightarrow e^+$ conversion offers a
complementary probe of lepton-number-violating physics.
