Transmission genetics with two loci

I'm going to construct a reduced version of a mating table to see how gamete frequencies change from one generation to the next. There are ten different two-locus genotypes (if we distinguish coupling, $A_1B_1/A_2B_2$, from repulsion, $A_1B_2/A_2B_1$, heterozygotes as we must for these purposes). So a full mating table would have 100 rows. If we assume all the conditions necessary for genotypes to be in Hardy-Weinberg proportions apply, however, we can get away with just calculating the frequency with which any one genotype will produce a particular gamete.³

		Gametes
Genotype	Frequency	$A_1B_1$	$A_1B_2$	$A_2B_1$	$A_2B_2$
$A_1B_1/A_1B_1$	$x_{11}^2$	1	0	0	0
$A_1B_1/A_1B_2$	$2x_{11}x_{12}$	$\frac{1}{2}$	$\frac{1}{2}$	0	0
$A_1B_1/A_2B_1$	$2x_{11}x_{21}$	$\frac{1}{2}$	0 $\frac{1}{2}$	0
$A_1B_1/A_2B_2$	$2x_{11}x_{22}$	$\frac{1-r}{2}$	$\frac{r}{2}$	$\frac{r}{2}$	$\frac{1-r}{2}$
$A_1B_2/A_1B_2$	$x_{12}^2$	0	1	0	0
$A_1B_2/A_2B_1$	$2x_{12}x_{21}$	$\frac{r}{2}$	$\frac{1-r}{2}$	$\frac{1-r}{2}$	$\frac{r}{2}$
$A_1B_2/A_2B_2$	$2x_{12}x_{22}$	0	$\frac{1}{2}$	0	$\frac{1}{2}$
$A_2B_1/A_2B_1$	$x_{21}^2$	0	0	1	0
$A_2B_1/A_2B_2$	$2x_{21}x_{22}$	0	0	$\frac{1}{2}$	$\frac{1}{2}$
$A_2B_2/A_2B_2$	$x_{22}^2$	0	0	0	1