Syntheme

A syntheme partitions six points into three duads, equivalently a perfect matching of the complete graph. There are $6!/(2^{3}3!)=15$ synthemes. Every duad belongs to three. Two edge-disjoint synthemes extend to a unique total of synthemes: their union is a six-cycle, whose complement has a unique factorization into three matchings.

A total is a set of five edge-disjoint synthemes covering all fifteen duads of a six-point set. There are six totals; each syntheme belongs to two and every two totals share exactly one syntheme. Each total contains exactly one syntheme containing any prescribed duad. Totals are one-factorizations of the complete graph.

Two disjoint synthemes form a six-cycle, whose complement is a triangular prism. The prism has a unique partition into three perfect matchings, so the pair extends uniquely to a total of synthemes. Eight synthemes are disjoint from a fixed syntheme; grouping them four per total yields two totals through it, and double counting gives six totals in all.

// Target: combinatorics.bigb