Combinatorics 5: X‑Chromosome Transitivity Principles

This is the fifth 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

 

Here are some strengthened transitive principles that apply when the sharing is on the X chromosome. What we can say depends on the the genders of the people involved.

Recall that men have one X chromosome and women have two X chromosomes. This is the reason for analyzing sharing on the X chromosome specially. In the case of a male, there is no difference between half-identical sharing and fully identical sharing when looking at regions on the X chromosome. In the case of a female, "sharing" refers to the usual half-identical sharing.

 
 
Transitive Principles for X-Chromosome Sharing

Anybody—Male—Anybody:
If person A shares with a male person on the X chromosome, and if that second person shares with a third person B, then A shares with B.

 
Male—Females—Male:
If male A shares with two or more females on the X chromosome, and if those females all share with male B, then either:

(a) More commonly — A and B share with each other;

or

(b) More rarely — All the females share fully identically on the region in question.

 
Male—Females—Female, or Female—Females—Male:
If person A shares with N females on the X chromosome, and if those N females all share with person B, and if A and B are of opposite sex, then either:

(a) More commonly — A and B share with each other;

or

(b) More rarely — At least N/2 (rounding up) of the N females share fully identically on the region in question.

 
The only other case is Female—Females—Female, but the best we can do there is the general version, applicable to all chromosomes.