Refining the relationship between homozygosity and the frequency of the most frequent allele

Recent work has established that for an arbitrary genetic locus with its number of alleles unspecified, the homozygosity of the locus confines the frequency of the most frequent allele within a narrow range, and vice versa. Here we extend beyond this limiting case by investigating the relationship between homozygosity and the frequency of the most frequent allele when the number of alleles at the locus is treated as known. Given the homozygosity of a locus with at most K alleles, we find that by taking into account the value of K, the width of the allowed range for the frequency of the most frequent allele decreases from 2/3 - π(2)/18 ≈ 0.1184 to 1/3 - 1/(3K) - {K/[3(K - 1)]} Σ(K)(k = 2) 1/k(2). We further show that properties of the relationship between homozygosity and the frequency of the most frequent allele in the unspecified-K case can be obtained from the specified-K case by taking limits as K → ∞. The results contribute to a greater understanding of the mathematical properties of fundamental statistics employed in population-genetic analysis.