The third moments of the site frequency spectrum

The analysis of patterns of segregating (i.e. polymorphic) sites in aligned sequences is routine in population genetics. Quantities of interest include the total number of segregating sites and the number of sites with mutations of different frequencies, the so-called site frequency spectrum. For neutrally evolving sequences, some classical results are available, including the expected value and variance of the spectrum in the Kingman coalescent model without recombination as calculated by FO_SCPLOWUC_SCPLOW (1995).\n\nIn this work, we use similar techniques to compute the third moments of the site frequency spectrum without recombination. We also account for the linkage pattern of mutations, yielding the full haplotype spectrum of three polymorphic sites. Based on these results, we derive analytical results for the bias of Tajimas D and other neutrality tests.\n\nAs an application, we obtain the second moments of the spectrum of linked sites, which is related to the neutral spectrum of chromosomal inversions and other structural variants. These moments can be used for the normalisation of new neutrality tests relying on these spectra.