In most of today's exactly solved classes of polyominoes, either all membersare convex (in some way), or all members are directed, or both. If the class isneither convex nor directed, the exact solution uses to be elusive. This paperis focused on polyominoes with hexagonal cells. Concretely, we deal withpolyominoes whose columns can have either one or two connected components.Those polyominoes (unlike the well-explored column-convex polyominoes) cannotbe exactly enumerated by any of the now existing methods. It is thereforeappropriate to introduce additional restrictions, thus obtaining solvablesubclasses. In our recent paper, published in this same journal, therestrictions just mentioned were semidirectedness and an upper bound on thesize of the gap within a column. In this paper, the semidirectednessrequirement is made looser. The result is that now the exactly solvedsubclasses are larger and have greater growth constants. These new polyominofamilies also have the advantage of being invariant under the reflection aboutthe vertical axis.
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