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An ancestral recombination graph records ancestral coalescences and genetic recombination events for sampled chromosomal material. Backwards in time, each pair of ancestral lineages merges at rate one in standard diploid coalescent units, while a lineage splits at a recombination breakpoint at rate $\rho /2$ times the genetic span of the material it carries. The breakpoint splits that lineage's ancestral material into two parental contributions. Waiting times are exponential with the sum of the current event rates. At each genomic position, retaining the ancestry at that position gives a marginal phylogenetic tree with the neutral Kingman's coalescent distribution. Different positions have dependent trees because they share parts of the graph.

// Target: biology.bigb

Association mapping localizes genetic influences on a phenotype by testing population genetic association with markers. Linkage disequilibrium allows a typed marker to tag a nearby causal variant; historical genetic recombination can narrow the associated interval. Population stratification can produce association without a local causal relation.

// Target: biology.bigb

Competition among independently arising beneficial lineages in a population with little or no genetic recombination. A favorable mutation can be lost because a competing lineage spreads more successfully.

Recombination of sequence fragments followed by reassembly, allowing parental sequence blocks to be combined. Homology, crossover distribution and epistasis affect the resulting diversity.^[ref]

Evolution is descent with inherited modification. Within a population, changes in allele frequencies provide a precise genetic description; over longer histories, branching descent connects populations through common descent. An individual growing larger is biological development, whereas an inherited variant becoming more frequent across generations is evolution. See UC Museum of Paleontology: the definition of evolution.

Mutation supplies new alleles, genetic recombination reshuffles inherited material, natural selection favors variants with greater reproductive contribution in a specified environment, and genetic drift changes frequencies through chance. Gene flow moves variants between populations. Consequently, an increasing allele frequency alone does not identify natural selection. See Clark, Douglas and Choi, Biology 2e: population genetics.

Evidence includes ordered fossils, shared inherited structures and similarities in DNA; phylogenetics uses such observations to infer relationships. A phylogenetic tree is an inferred historical account, with uncertainty about particular branches. See Clark, Douglas and Choi, Biology 2e: evidence of evolution. Evidence also comes from experiments: Blount, Borland and Lenski reported an Escherichia coli lineage able to use citrate with oxygen present by 31,500 generations, and tested historical dependence by replaying evolution from stored ancestors. See Blount, Borland and Lenski, 2008: original experimental report. Such observations constrain explanations; no single frequency equation captures every evolutionary process.

The proposal that genetic recombination can accelerate adaptation by combining beneficial alleles that arise in different lineages. Recombination also produces other combinations, so the benefit depends on the inherited associations and ensuing natural selection.

Under an infinite sites mutation model, two variable sites on one unrecombined rooted tree have nested or disjoint derived-allele clades. At most three of the four biallelic haplotypes can then occur. Observation of all four haplotypes proves that the single-tree, single-mutation-per-site model cannot hold: some ancestral genetic recombination is necessary if recurrent mutation is excluded. The test gives no proof of absence of recombination when only three haplotypes occur.

// Target: biology.bigb

A genetic map turns observations of inherited haplotypes into an ordered account of genetic loci along a chromosome. Its distances describe genetic recombination, which makes the map useful for locating a genetic locus relative to known genetic markers. A genetic map measures inheritance separation; separation measured in base pairs describes positions along DNA. Recombination hotspots help explain why equal numbers of base pairs need not give equal genetic map distances. See T. A. Brown, Genomes, 2nd edition (2002), chapter 5: Mapping Genomes.

Write $d$ for the expected value of the number of chromosomal crossovers encountered along one transmitted chromatid between the chosen genetic loci, measured in morgans; $100d$ is the distance in centimorgans. This counts exchanges along that transmitted chromatid, rather than all exchanges in the four-chromatid meiotic bundle. For a short interval with very few multiple exchanges, the recombination fraction satisfies $r\approx d$: about $1\%$ recombinant gametes then corresponds to about one centimorgan.

For a longer interval, hidden even numbers of chromosomal crossovers make that approximation inaccurate. Under the Haldane mapping function, $d=0.2$ morgans gives $r\approx 0.164840$; conversely $r=0.2$ gives $d\approx 0.255413$ morgans, or $25.5413$ centimorgans. Applying the inverse formula to an already specified genetic map distance would confuse a count with a probability.

Genetic map distances add along ordered adjacent intervals because the total exchange count is the sum of its interval counts. For independent random variables describing counts in disjoint intervals of the Haldane mapping function model, the endpoint recombination fraction instead obeysExactly one interval must change parental origin. Thus $r_1=r_2=0.2$ gives $r_{12}=0.32$, although the two genetic map distances add to approximately $0.510826$ morgans. The simple composition formula requires the stated independence; crossover interference can invalidate it.

Genetic recombination rearranges inherited genetic material into new combinations. In the meiosis example considered here, chromosomal crossover exchanges segments between nonsister chromatids of homologous chromosomes. A transmitted chromatid can consequently contain DNA from both parental homologues. This produces new haplotypes without requiring a new mutation at either scored genetic locus. See Meiosis.

For phase-known genotype $AB/ab$, a single chromosomal crossover between the selected genetic loci can yield $Ab$ and $aB$ on the participating chromatids; the other two chromatids can remain $AB$ and $ab$. Thus an exchange in the four-chromatid bundle does not mean every resulting gamete is recombinant. Only a chromosomal crossover encountered along the transmitted chromatid within the selected interval affects its two-locus score.

Follow parental origin along the transmitted chromatid from the first genetic locus to the second. Each exchange switches that origin. An odd number of such chromosomal crossovers changes the endpoint combination; an even number restores parental phase. Two exchanges can therefore leave the scored haplotype as $AB$ even though an internal segment came from the other parental homologous chromosome. Recombinant status is relative to the chosen genetic loci, not a complete record of exchanges along the chromosome.

This integer parity distinction connects the physical process of genetic recombination to the observed recombination fraction. It also explains why a long genetic map interval can accumulate many exchanges while its endpoint recombination fraction approaches $1/2$ in the Haldane mapping function model.

Locating genetic markers or other features within a genome. Physical positions use DNA sequence distances, while linkage maps use genetic recombination frequencies.

A haplotype is the set of alleles on one chromosome across specified genetic loci. Closely linked markers without intervening genetic recombination are transmitted as one haplotype; they are not independent observations of inheritance.

A haplotype block is a chromosomal region with limited historical genetic recombination, strong local linkage disequilibrium and comparatively few common haplotypes. Its definition is operational and population-dependent; it does not mean that recombination never occurs within it.

// Target: biology.bigb

Reduced efficacy of natural selection caused by associations between selected genetic loci in finite populations. genetic recombination can relieve some of this interference by changing the associations.

Reduced survival or fertility appearing in later-generation or backcross descendants of hybrids, even when first-generation offspring are fertile. genetic recombination can expose harmful combinations previously masked in the first generation.

An identical-by-descent segment is a chromosomal tract inherited from the same ancestral chromosome without an intervening breakpoint in the tracked ancestry. With a specified total number $m$ of independent meioses and a uniform recombination map, the probability that a fixed tract of genetic length $d$ survives without genetic recombination is approximately $e^{-md}$. Recent common ancestry therefore tends to give longer shared tracts. Lengths of tracts selected by covering a random point are length-biased relative to tracts sampled by their boundaries.