# Is there a vertical measure of appearance of a genotype over generations?

Is there some kind of a vertical measure of appearance of a genotype $$g$$ throughout ancestry, i.e. a measure of how many ancestors of $$g$$ had genotype $$g$$. Like for example in saying that on average each member of a population $$P$$ have 60% of its ancestors having genotype $$g$$ over 20 prior generations.

A clarifying example, lets take some genetic trait like color, suppose we have a population $$P$$, and suppose we know all of the ancestry of each individual $$x$$ in that population up to say six generations, i.e. we know specifically the color of each mother and father of $$x$$, the grandmothers and grandfathers of $$x$$, the first great grandmothers and the first great grandfathers of $$x$$, up to .. the fourth great grandparents, call the set of all those as $$Ancestry^6(x)$$, lets denote the set of all individuals of $$Ancestry^6(x)$$ that are $$a$$ colored as $$Ancestry^6_a$$, let's denote the total number of members of any set $$y$$ by $$|y|$$,

Now what I'm looking at is the proportion $$p^6_a(x)$$ of all $$a$$ colored members of the 6 generations ancestry of $$x$$, from that ancestry.

$$p^6_a(x)=|Ancestry^6_a(x)|/|Ancestry^6(x)|=|Ancestry^6_a(x)|/(2+2^2+..+2^6)$$

Now for population $$P$$ we'll look for average $$p^6_a(x)$$: $$x \in P$$, symbolized as $$\mu^P(p^6_a)$$ which is $$\mu^P(p^6_a) = \big{(}\sum^P p^6_a(x)\big{)}/|P|$$

In general the measure is $$\mu^P(p^m_a)$$, where $$P$$ is the population, $$m$$ the total number of generations in the ancestry of each member of $$P$$, $$a$$ is the genotype, and $$\mu$$ signify average, and $$p$$ for proportion. So $$\mu^P(p^m_a)$$ is: the average proportion of genotype $$a$$ in ancestry of each member of $$P$$ up to $$m$$ generations.

Question: is that measure directly proportional to fitness?

The idea is that if genotype $$a$$ is more fit than others, then it would appear more frequently in the ancestry of each member in the population.

• I don't really understand the statistic that you are trying to describe but it sounds like you might want to read about "coalescent theory" (the usage of branching patterns to describe patterns of ancestry among individuals) and the "folder/unfolded site frequency spectrum" (SFS; frequency distribution of allele frequency in a population). The SFS is one of the many statistics for which we can get theoretical expectation from coalescent theory (see Watterson's estimator function maybe). Apr 2, 2019 at 0:02
• Remi.b, OK I've added a clarifying example, that defines it exactly. Apr 2, 2019 at 7:10