  The invariants of rank 2 Joyce-Song semistable pairs over a Calabi-Yau
threefold were computed in arXiv:1101.2252, using the wall-crossing formula of
Joyce-Song and Kontsevich-Soibelman. Such wall-crossing computations often
depend on the combinatorial properties of certain elements of a Hall-algebra
(these are the stack functions defined by Joyce). These combinatorial
computations become immediately complicated and hard to carry out, when
studying higher rank stable pairs with rank$>2$. The main purpose of this
article is to introduce an independent approach to computation of rank 2 stable
pair invariants, without applying the wallcrossing formula and rather by
stratifying their corresponding moduli space and directly computing the
weighted Euler characteristics of the strata. This approach may similarly be
used to avoid complex combinatorial wallcrossing calculations in rank$>2$
cases.
