To derive haplotype frequencies at time *t* as functions of haplotype frequencies at time 0, we used the Bennett decomposition of haplotype frequencies [19] and the work of [20].

Let

*A*
_{
n
} denote a set of

*n* alleles at

*n* different loci,

*A*
_{
n
} = {

*a*
_{1},

*a*
_{2}, ...,

*a*
_{
n
}}. Let

*D*
_{
n
}(

*A*
_{
n
} ,

*t*) be the n-loci linkage disequilibrium of

*A*
_{
n
} alleles at time

*t* defined by [

19] such that, in an infinitely large population, under random mating and meiosis

where *ρ*{*A*
_{
n
} } is the probability of no recombination across loci belonging to *A*
_{
n
} .

Assuming no interference between loci leads to

where *c*
_{
i, i'
} is the recombination rate between loci *i* and *i'*.

Let

(

*t*) be the frequency of the haplotype carrying the alleles in

*A*
_{
n
} at time

*t*. Then by definition

where the coefficients
are constants obtained by recursion [20], and *p* = {⋃_{
i
}
*A*
_{
ni
} = *A*
_{
n
}} denotes a partition of *A*
_{
n
}. For example, for *n* = 3 there are 5 partitions namely {*a*
_{1}, *a*
_{2}, *a*
_{3}}, {{*a*
_{1}, *a*
_{2}}⋃{*a*
_{3}}}, {{*a*
_{1}, *a*
_{3}}⋃{*a*
_{2}}}, {{*a*
_{2}, *a*
_{3}}⋃{*a*
_{1}}} and {{*a*
_{1}}⋃{*a*
_{2}}⋃{*a*
_{3}}}.

When

*n* equals two and three, [

20] proved that the

are all equal to one. But when

*n* ≥ 4, some

are not equal to one even if we assume no interference between loci. For example, for the partition {{

*a*
_{1},

*a*
_{4}}⋃{

*a*
_{2},

*a*
_{3}}} with four loci, [

20] proved that

which does not reduce to unity, except for unlinked loci. This means that, for *n* ≥ 4, the Bennett disequilibria are different from disequilibria defined by [27–29] since these authors imposed
= 1 in formula (2). However, the Bennett disequilibria are the only multilocus linkage disequilibrium measures that decay geometrically with time.

Let

*n* be odd and composed of (

*n* − 1)/2 left and right markers surrounding a putative causal locus. Assume that at time 0 all the Bennett disequilibria between markers are null, i.e. markers were in equilibrium when the causal mutation appeared. Formula

(1) states that marker disequilibria are null throughout the population history. Moreover, all the terms not equal to zero in the formula

(2), applied to the frequency of markers and the mutated locus haplotypes, have a

constant equal to one. Partitions that do not involve marker disequilibria are such that

where the causal locus is in the set

*A*
_{
p
} and

*k* = 0 means

*A*
_{
p
} =

*A*
_{
n
}. Since those partitions are composed of singletons and a single subset of

*A*
_{
n
},

= 1 (formula 4.14 in [

20]), then we get

where #

*A*
_{
p
}denotes the cardinal of set

*A*
_{
p
}. We finish the calculation by using the reverse formula of

*D*
_{#Ap
}(

*A*
_{
p
}, 0) as a function of haplotype frequencies at time 0, which in this case can be obtained easily using recursion based on the following equation

In a finite population, formulae developed in an infinite population, can be transformed using the expectation of multi-locus disequilibria and haplotype frequencies, and taking only the first order development of these expectations as the population size extends to infinity. We then get

where ≃ means asymptotically equivalent.

Equalities of first order developments are based on the fact that products of expectations are asymptotically equal to expectations of products. These equalities can also be found using the work of [27].