We introduce an associated version of the binomial inversion for unified Stirling numbers defined by Hsu and Shiue. This naturally appears when we count the number of subspaces generated by subsets of a root system. We count such subspaces of any dimension by using associated unified Stirling numbers, and then we will also give a combinatorial interpretation of our inversion formula. In particular, the well-known explicit formula for classical Stirling numbers of the second kind can be understood as a special case of our formula.
"Associated binomial inversion for unified Stirling numbers and counting subspaces generated by subsets of a root system." Tsukuba J. Math. 42 (1) 97 - 125, July 2018. https://doi.org/10.21099/tkbjm/1541559652