Relationship between Ploidy, Alleles and Genotypes

Introduction

The VCF format encodes genotypes by the index of the enumeration of genotypes give a ploidy number and alleles. This allows for direct access to a genotype value within an array particularly when one works with genotype likelihoods.

Motivation

Plants species exhibit a diverse number of ploidy, for example, the strawberry is an octoploid and the pear is a triploid.

While there are explicit functions that could be googled for handling haploid and diploidy cases. It seems to be difficult to find the closed forms for the general case. This wiki fills in that need.

The number of genotypes given a ploidy and alleles

${\begin{aligned}F(P,A)={\begin{cases}{\binom {P+A-1}{A-1}}&,A<P\\\sum _{i=1}^{P-1}{\binom {A}{i}}{\binom {P-1}{i-1}}+{\binom {A}{P}}&,A\geq P\end{cases}}\end{aligned}}$

where P is the ploidy number and A is the number of alleles.

The indexing of genotypes given a ploidy and alleles

${\begin{aligned}G(a_{1},..,a_{p})={\begin{cases}a_{1}&,p=1\\F(a_{p},p)+G(a_{1},..,a_{p-1})&,p\neq 1\end{cases}}\end{aligned}}$

where a_1, a_2 .... are the alleles in numeric encoding (0 to A-1) and are ordered (AB, ABCCCC). For example ACB is not ordered.

Simple cases

Ploidy	Alleles	Genotypes	Index
1	A	$F(1,A)=A$	$G(a_{1})=a_{1}$
2	A	$F(2,A)=A+{\binom {A}{2}}$	$G(a_{1},a_{2})={\binom {a_{2}+1}{2}}+a_{1}$