The traditional rule for succession in a monarchy is to pass from father to son. Much of King Henry VIII’s spousal folly was over his anxiety for producing an heir. However, if the basis of being a successful ruler has a genetic component then this would be the least optimal way to run an empire. For diploid sexually reproducing organisms, such as humans, the offspring inherits equal numbers of chromosomes from both parents and classically the genetic relationship or kinship coefficient between parent and child is assigned the value of 1/2. However, there is a crucially important asymmetry in that males are heterozygous in the sex chromosomes, i.e. they inherit an X chromosome from their mothers and a Y from their fathers, while females are homozygous, inheriting an X from both. Now the X is about 100 million base pairs longer than the Y, which accounts for about 2 percent of the (father’s) genome (counting chromosomes separately). Additionally, given that everyone has at least one X while only males have a Y, the Y cannot contain genes that are crucial for survival and in fact there are much fewer genes on the Y than the X (~800 vs ~50). The Y has been shrinking in mammals over time and there is a debate about its importance and eventual fate (e.g. see here).
We can compute the sex chromosome adjusted genetic correlation coefficients between parents and children. Let the father’s genetic content be , where is the genetic content passed to sons (half of the autosomes plus the Y chromosome) and is that passed to daughters (half of the autosomes plus the X) and similarly . The son genetic content is then and daughter is . We can treat F and M as a string of random variables with variance 1/(length of mother’s genome) and assuming that the genetic correlation between fathers and mothers is zero (i.e. no inbreeding and no assortative mating) then the correlation coefficient between father and son is
Now, if you assume that genetic content is homogeneous among all chromosomes then that would mean that the genetic material that fathers pass on to sons is 0.48 of the total and thus while , , and implying that , , , . Hence, parents are more correlated with their children of the opposite sex and fathers are least correlated with their sons. These numbers also probably underestimate the asymmetry. If genetic relationship is the most important factor for royal succession then a dynasty based on opposite sex succession will be more logical than the father to son model.