Sortabilities of partition properties in multi-dimensional parameter spaces

The theory of sortability of a partition property was introduced to prove the existence of an optimal partition satisfying the property for optimal partition problems over single-parameter space, and then extended to multi-dimensional parameter spaces. For each partition property of interest, almost all levels of sortabilities have been obtained; however, the part-specific-sortabilities are hard to be determined for many properties. In this paper, we establish a rule to generate examples that reveal the non-part-specific-sortabilities of these properties for almost all cases. Such a rule also has potential of generating more concise examples to support known results.