Given an integer g, g > 1, an integer w, -1 < w <g - 1, and a set of g distinct numbers, c_1, ..., c_g, we present a conjectured identity for Stirling numbers of the first kind. We have proven all the equalities in case g < 7; and for the case g = 7, provided w < 4. These expressions arise from an aspect of the study of the dimer-monomer problem on regular graphs.