We developed and applied a data inversion routine to determine the number based size-mass distribution function (the two dimensional distribution function) from tandem differential mobility analyzer-aerosol particle mass analyzer (DMA-APM) measurements. The two dimensional distribution function is expressed in units of particle number concentration per unit mobility diameter per unit particle mass. It can be used to directly calculate the number based size distribution (commonly determined using DMA measurements) or the mass based size distribution (commonly inferred from impactor measurements). The inversion routine utilizes the Twomey-Markowski algorithm and is applied in this study to DMA-APM measurements of sodium chloride, cesium iodide, and ammonium sulfate particles in the 30-200 nm mobility diameter range, as well as acetylene flame generated soot aggregates in the 40-350 nm range. To utilize the inversion routine, the APM transfer function must be known a priori. Here it is computed using a modified version of the Ehara (uniform flow) model, with a transmission correction factor inferred from measurements. For the three examined salt particle types, visual representation of the two dimensional distribution function reveals that at a given mobility diameter, particles have very narrow mass distributions, with the peak masses in good agreement with predictions based on bulk salt densities. However, for soot particles, extremely broad distributions are observed. Soot measurements are compared to predictions for quasifractal aggregates in the transition regime; this comparison suggests that aggregates with fractal dimensions ranging from 1.4 to 2.5 are all generated in the same system. Finally, we determine the two-dimensional distribution function for a mixture of ammonium sulfate and soot particles, demonstrating that these two particle populations are separable from one another via mobility-mass analysis. (C) 2015 Elsevier Ltd. All rights reserved.