The intention is to show the relationship between the various designs when they are generated cyclically.
Starting with a BIBD with a block size of 2k, here 6, the first table represents the differences between the elements of an initial block of the design. Taken over the full set of initial blocks, these differences must comprise an integral number (λ) of the full set of possible differences (ignoring complications such as partial cycles and invariant elements). Let us say this has one condition on the elements.
The next table represents the differences arising in a NBIBD of Preece type, with sub-blocks of size k, here 3. The set of differences within the sub-blocks, combined, comprise λ2 occurrences of all possible differences; suppose these to be in two sub-tables of size k on the main diagonal. Those elsewhere (not in the main diagonal sub-tables) must comprise λ1 - λ2 occurrences. This represents two conditions on the elements of the blocks.
The third table represents an NBIBD of the more general type, in which each sub-block must represent a BIBD. The differences arising within the two sub-blocks, separately, must each comprise λ2 occurrences: thus the two sub-tables on the main diagonal must be considered separately. The combined set of elements in the off-diagonal sub-tables comprise another integral number of occurrences, say λ1 - 2λ2 occurrences. This represents three conditions.
In the fourth table, a correlated block design with blocks of size 3, assume the first sub-block corresponds to the first set of treatments, and the second sub-block to the second set. The differences between the treatment sets must comprise an integral number of occurrences of all the differences. Thus the two off-diagonal sub-squares must be considered separately, but one is just the negation of the other, so let us say (provisionally) this represents four conditions.
In a pergola, there must be a transversal through each of the off-diagonal sub-squares; this would give five conditions (since one is the negation of the other).
Finally, there is the PMD in blocks of size 6: this must have 5 transversals, representing the differences 1-apart, 2-apart, etc. But two of these are just negations of others, so there are just four conditions, on this analysis. In the general case, the number of conditions on a PMD increases with block size, whereas it is fixed with the others.
It must be said that this approach cannot be the whole story, since pergolas exist where the equivalent PMDs do not exist. In particular, OBIBDs have twice the number of blocks of the corresponding PMD. Factors such as resolvability also affect matters.
In the tables, following, the elements on the leading diagonal are all zeros, so are to be ignored.
Let us say this has one condition imposed on the elements.
This has two conditions, by the same principle.
And this has three.
The green elements above are just the negations of the purple, so are not imposing any extra conditions. There are just 4 distinct conditions.
The brown and green elements above are just the negations of the yellow and purple, so are not imposing any extra conditions. There are just 5 distinct conditions.
This would seem to have just 4 conditions (since 2 of the transversals are negations of 2 others). But in general, PMDs are stricter than pergolas, having the number of conditions increasing with block size.