Donovan Young on Nostr: When we have four sets, there is no longer a unique solution to the resulting linear ...

When we have four sets, there is no longer a unique solution to the resulting linear system.

There are now six sorts of edges, see the attached figure. The linear system is:

\(p_1+p_4+p_5=k_1\)
\(p_1+p_2+p_6=k_2\)
\(p_2+p_3+p_5=k_3\)
\(p_3+p_4+p_6=k_4\)

This is four equations in six unknowns, and so, assuming there are perfect matchings to count, there is a two-parameter family of them.