The partial isometry homology groups $H_n$ defined in Power [1] and a related chain complex homology $CH_*$ are calculated for various triangular operator algebras, including the disc algebra. These invariants are closely connected with $K$-theory. Simplicial homotopy reductions are used to identify both $H_n$ and $CH_n$ for the lexicographic products $A(G) \star A$ with $A(G)$ a digraph algebra and $A$ a triangular subalgebra of the Cuntz algebra $O_m$. Specifically $H_n (A(G) \star A) = H_n (\Delta (G)) \otimes_{{\mathbb Z}} K_0 (C^* (A))$ and $CH_n (A(G) \star A) $ is the simplicial homology group $ H_n (\Delta (G) ; K_0 (C^* (A)))$ with coefficients in $K_0 (C^* (A))$. [1] S. C. Power, {\em Homology for operator algebras\/} II: {\em Stable homology for non-self-adjoint algebras}, J. Functional Anal., {\bf10} (1996) 233--269.