Colored q-Stirling and q-Lah numbers: A new view continued

Cai and Readdy proposed a new framework for studying the q-analogue f(q) of a combinatorial structure S. Specifically, the aim is to identify two statistics over S and a proper subset S^′ of S such that f(q) represents the q-(1+q)-expansion over S^′, and to explore the poset and topological interpretations of this expansion. Cai and Readdy provided comprehensive profiles for classical Stirling numbers of both kinds within this framework. In this work, we extend Cai and Readdy's results to colored q-Stirling numbers of both kinds, as well as colored q-Lah numbers. We also briefly discuss q-Stirling and q-Lah numbers of type D.