Combinatorics 2: Non‑transitivity

This is the second in a series of 7 posts on sharing combinatorics:

Part 1: Combinatorial Principles for DNA Sharing

Part 2: Non-transitivity

Part 3: A Genetic Pigeonhole Principle

Part 4: Transitivity Principles

Part 5: X-Chromosome Transitivity Principles

Part 6: Mutual Sharing Principle

Part 7: Exceptions

 

Sharing isn't transitive, as I've already mentioned: it's quite possible for Person A to share a segment with Person B, and for Person B to share that same segment with Person C, but for Person A not to share with Person C.

For example, Person A and Person B could have the same bases 1,000,000 to 2,000,000 on one copy each of Chromosome 1. And Person B and Person C could also share the same bases 1,000,000 to 2,000,000 on one copy each of Chromosome 1. But we could be talking about the two different Chromosome 1's in Person B:

As you can see, Person A doesn't share on that segment with Person C — even though A shares with B and B shares with C.

After all, if A is related to B's mother, and C is related to B's father, there' s no reason to expect that Person A and Person C should be related to each other.

We'll see that, even though sharing isn't transitive, there are some combinatorial laws that do apply to genome sharing, including a transitive-like law.