As Prof Raabe mentioned, in terms of theoretical approach, the combined use of ab-initio and thermodynamic calculations is the quite promising way to know stacking fault energy of austenitic steels. In fact, the experimental data of stacking fault energy has been reported to decrease first and decrease with increasing Mn content. This tendency is also successfully simulated by the calculations.
From the viewpoint of experiments, the stacking fault energy can be measured by XRD peak profile analysis, curvature measurement of SF nodes by TEM, and direct measurement of partial dislocation spacing by TEM.
Answering to the most important question "the effect of Mn" is NOT easy. This is because stacking fault energy of austenitic steels with Mn is strongly affected by magnetism, interfacial energy, and probably interaction parameter with other elements. What I can say for the effect of Mn is "the effect of Mn on SFE is not monotonic". At least, addition of high concentration of Mn (>22Mn) seems to increase SFE.