I'm pretty rusty on my biology, but I would think each of the four "parents" would have to contribute genetically. The way it works would, clearly, have significant differences from our own. The equivalent of spermogenesis and oogenesis would spilt in fourths, instead of halves, for one. I'm still trying to hash out in my head what the equivalent of XX and XY would be for a four-gendered species, how that would work.
Of course, presuming you have four contributers, then that chromosome would have four components, and you could still do it with only two types. For example:
Chan: XYYY
Shen: XXYY
Zhen: XXXY
Thaan: XXXX
Now, a flaw I can see in this methodology (which might be part of why the Andorians were dying out) is (unless my math is bad) you're going to get, on average, more Shen and Zhen then Thaan and Chan. And therefore your number of viable mating groups drops, and thus the population drops.
(And this might be, also, what happened on Andor over the eons: most of the other four-gendered species died out due to this imbalance.)
Are those completely random combinations, with X and Y standing for nothing in particular? If not, I am even more confused. Why would a thaan have a combination more similar to a zhen than to a chan?
The "X" and "Y" in this case refer to two hypothetical sex-determining chromosomes, which form a base-quad in the Andorian-equivalent of our 23rd chromosome. "Ш" and "Ѳ" could work just as well. My point was to demonstrate that with base-quads, only two types are necessary to create four genders. The exact quad-combination to gender is just a suggestion on my part, and if someone has one that works better, by all means.
Well, if
thaan and
chan are supposed to correspond to "male", and
zhen and
shen to "female", with chan and shen the more androgynous ones, unless Memory Alpha is lying to me, it seems to me that it would make a lot more sense if it went something like this:
zhen: XXXX
shen: XXXY
chan: XXYY
thaan: XYYY
Kind of like in human genetics where the people who happen to be
XXY are androgynous compared to most XY males. It seems to me that a
shen should have a combination more similar to a
zhen than to a
thaan, and that a
thaan should have a combination more similar to that of a
chan than to a
shen. Regardless of the whole androgyny thing, "female"
zhen should have a combination more similar to "female"
shen than to the other two, "male" sexes.
Your main point, however, stands - I did the maths, and unless I made a mistake in calculations, the statistical probability is that there would be just 3 thaans and 3 zhens to 13 chans and 13 shens. The four person marriage doesn't really make sense in that context, what do all those other chans and shens do? Or are they just free to have recreational sex and not participate in the reproduction?
But then, as you said, this might be the reason why they're dying out. Still, I don't see how they think it could be fixed with a four person marriage.