Notes:
This is a real SFO-BOS round trip example with numbers, constrained to a particular American Airlines route (SFO to BOS through ORD, then after a Saturday night back from BOS to SFO through DFW) with travel each way limited to one calendar day. There are a total of 25,401,415 valid solutions from a space of more than 10,000,000,000 combinations of flights and fares and priceable units.
Exploring the diagram, from SFO to ORD on the travel date AA offers 5 flights and publishes 36 fares in the market. After testing those fare rules that restrict fare component flights and times, there are a total of 85 possible SFO to ORD fare components (from a space of 5 * 36 = 180). Similarly, from SFO to BOS the 5 SFO to ORD flights combine with the 7 ORD to BOS flights to produce 19 flight combinations (instead of 35, because of time constraints). AA publishes 32 SFO-BOS fares, but the 19 flight combinations * 32 fares only produce 109 fare components after checking appropriate rules.
When constructing priceable units, the 112 outbound ORD to BOS fare components combine with the 87 return BOS to DFW fare components to produce 8,169 open jaw priceable units. Again this is less than 112 * 87 because fare rules limit combinations. There are 14 possible priceable unit geometries, and 13 ways to use the geometries to cover all four flights.
Putting together all the possible ways to combine all the possible priceable units into a complete ticket, there are 25,401,415 solutions. This is just for this particular airline and route – it represents a very small portion of the search space that an engine would need to consider for an unrestricted SFO to BOS round trip journey.