Lattice equations arising from discrete Painlev\'e systems (I): $(A_2+A_1{)}^{(1)}$ and $(A_1+A_1^{\mathrm{\prime }}{)}^{(1)}$ cases

We introduce the concept of $\omega$-lattice, constructed from $\tau$ functions of Painlev'e systems, on which quad-equations of ABS type appear. In particular, we consider the $A_5^{(1)}$- and $A_6^{(1)}$-surface $q$-Painlev'e systems corresponding affine Weyl group symmetries are of $(A_2+A_1{)}^{(1)}$- and $(A_1+A_1{)}^{(1)}$-types, respectively.