# Continuous Time Markov Chain

This work is in connection with the CoalHMM, coalescent hidden Markov model. CoalHMM is a statistical framework of inferring population genetic parameters. In the field of Coalescence theory, we give probabilities to all the possible gene genealogies that could have created the varication seen in the present day samples. The typical description of the coalescence model is as a continuous time Markov process running backwards in time, describing the various events that could have occurred in the past.

## CTMC Construction

For computing transition probabilities, we explicitly construct the state space of the two-loci coalescence process. This state space describes all possible histories within one time slice. This approach is used in several earlier papers [1, 2, 3, 4]. We represent lineages at a single nucleotide as sets. The sets [1] and [2] denote sequences 1 and 2 before they have found a common ancestor while [1, 2] denotes a lineage ancestral to both. We then model two neighboring nucleotides as pairs of such states; e.g. ([1, 2] , [1]) denote a lineage in which the left nucleotide has found a common ancestor between samples 1 and 2 and is linked on the right to a nucleotide from sequence 1, which has not found a common ancestor with sequence 2. To assign lineages to species, we pair them again and let (1, (l, r)) denote that lineage (l, r) is in population 1. A state in the CTMC corresponds to a set of such lineages assigned to species.

## CTMC Projection

Time slices work together when they have the same CTMC state space. When the state spaces are different, however, we need projection matrices to move samples from one state space to another. We obtain this state mapping by resetting population labels in the set representations of CTMC states. For example, [(1, ([1], [1])), (2, ([2], [2]))] in a two population period is mapped to [(0, ([1], [1])), (0, ([2], [2]))] in a single population period. For example, if we were to connect the two-sample isolation CTMC with the two-sample single-population CTMC, shown in the figure above, we would need the following projection matrix.

```      | 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14
--+--------------------------------------------
0 | 1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
1 | 0  0  0  0  1  0  0  0  0  0  0  0  0  0  0
2 | 0  1  0  0  0  0  0  0  0  0  0  0  0  0  0
3 | 0  0  0  0  0  1  0  0  0  0  0  0  0  0  0
```

## Two Population Isolation CTMC

For two samples in two isolated populations, there are 4 states as shown in the following figure. The label for these two populations are 1 and 2.

```    [ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 0 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [2] ]]: 1 ],
[ [[ 1, [1], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 2 ],
[ [[ 1, [1], [1] ], [ 2, [2], [2] ]]: 3 ]
```

Copied below is the corresponding rate matrix, where R is the recombination rate, C is the coalescent rate. The row sum is zero so the diagonal values are the negated sum of all other values on each row.

```    	0	1	2	3
0	-	c2	c1	0
1	r	-	0	c1
2	r	0	-	c2
3	0	r	r	-
```

## Single Population CTMC

For two samples in one population, there are 15 states as shown in the following figure. The label for this single population is 0.

```    [ [[ 0, [1], [] ], [ 0, [2], [] ], [ 0, [], [1] ], [ 0, [], [2] ]]: 0 ],
[ [[ 0, [1], [2] ], [ 0, [2], [] ], [ 0, [], [1] ]]: 1 ],
[ [[ 0, [1], [1] ], [ 0, [2], [] ], [ 0, [], [2] ]]: 2 ],
[ [[ 0, [1], [] ], [ 0, [2], [1] ], [ 0, [], [2] ]]: 3 ],
[ [[ 0, [1], [] ], [ 0, [2], [2] ], [ 0, [], [1] ]]: 4 ],
[ [[ 0, [1], [2] ], [ 0, [2], [1] ]]: 5 ],
[ [[ 0, [1], [1] ], [ 0, [2], [2] ]]: 6 ],
[ [[ 0, [1,2], [] ], [ 0, [], [1] ], [ 0, [], [2] ]]: 7 ],
[ [[ 0, [1,2], [2] ], [ 0, [], [1] ]]: 8 ],
[ [[ 0, [1,2], [1] ], [ 0, [], [2] ]]: 9 ],
[ [[ 0, [1], [] ], [ 0, [2], [] ], [ 0, [], [1,2] ]]: 10 ],
[ [[ 0, [1], [] ], [ 0, [2], [1,2] ]]: 11 ],
[ [[ 0, [1], [1,2] ], [ 0, [2], [] ]]: 12 ],
[ [[ 0, [1,2], [] ], [ 0, [], [1,2] ]]: 13 ],
[ [[ 0, [1,2], [1,2] ]]: 14 ]
```

Copied below is the corresponding rate matrix, where R is the recombination rate, C is the coalescent rate. The row sum is zero so the diagonal values are the negated sum of all other values on each row.

```    	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14
0	-	c1	c1	c1	c1	0	0	c1	0	0	c1	0	0	0	0
1	r	-	0	0	0	c1	0	0	c1	0	0	0	c1	0	0
2	r	0	-	0	0	0	c1	0	0	c1	0	0	c1	0	0
3	r	0	0	-	0	c1	0	0	0	c1	0	c1	0	0	0
4	r	0	0	0	-	0	c1	0	c1	0	0	c1	0	0	0
5	0	r	0	r	0	-	0	0	0	0	0	0	0	0	c1
6	0	0	r	0	r	0	-	0	0	0	0	0	0	0	c1
7	0	0	0	0	0	0	0	-	c1	c1	0	0	0	c1	0
8	0	0	0	0	0	0	0	r	-	0	0	0	0	0	c1
9	0	0	0	0	0	0	0	r	0	-	0	0	0	0	c1
10	0	0	0	0	0	0	0	0	0	0	-	c1	c1	c1	0
11	0	0	0	0	0	0	0	0	0	0	r	-	0	0	c1
12	0	0	0	0	0	0	0	0	0	0	r	0	-	0	c1
13	0	0	0	0	0	0	0	0	0	0	0	0	0	-	c1
14	0	0	0	0	0	0	0	0	0	0	0	0	0	r	-
```

## Three Population with Migration in Two CTMC

For two samples in three populations where migration is allowed between two populations, there are 12 states as shown in the following figure. The label for these three populations are 1, 2, and 3, and migration happens between 2 and 3.

```    [ [[ 1, [1], [] ], [ 1, [], [1] ], [ 3, [2], [] ], [ 3, [], [2] ]]: 0 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 1 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [], [2] ], [ 3, [2], [] ]]: 2 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [] ], [ 3, [], [2] ]]: 3 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 3, [2], [2] ]]: 4 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [2] ]]: 5 ],
[ [[ 1, [1], [1] ], [ 3, [2], [] ], [ 3, [], [2] ]]: 6 ],
[ [[ 1, [1], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 7 ],
[ [[ 1, [1], [1] ], [ 2, [], [2] ], [ 3, [2], [] ]]: 8 ],
[ [[ 1, [1], [1] ], [ 2, [2], [] ], [ 3, [], [2] ]]: 9 ],
[ [[ 1, [1], [1] ], [ 3, [2], [2] ]]: 10 ],
[ [[ 1, [1], [1] ], [ 2, [2], [2] ]]: 11 ]
```

Copied below is the corresponding rate matrix, where R is the recombination rate, C is the coalescent rate. The row sum is zero so the diagonal values are the negated sum of all other values on each row.

```    	0	1	2	3	4	5	6	7	8	9	10	11
0	-	0	m32	m32	c3	0	c1	0	0	0	0	0
1	0	-	m23	m23	0	c2	0	c1	0	0	0	0
2	m23	m32	-	0	0	0	0	0	c1	0	0	0
3	m23	m32	0	-	0	0	0	0	0	c1	0	0
4	r	0	0	0	-	m32	0	0	0	0	c1	0
5	0	r	0	0	m23	-	0	0	0	0	0	c1
6	r	0	0	0	0	0	-	0	m32	m32	c3	0
7	0	r	0	0	0	0	0	-	m23	m23	0	c2
8	0	0	r	0	0	0	m23	m32	-	0	0	0
9	0	0	0	r	0	0	m23	m32	0	-	0	0
10	0	0	0	0	r	0	r	0	0	0	-	m32
11	0	0	0	0	0	r	0	r	0	0	m23	-
```

## Two Population with One-direction Migration CTMC

For two samples in two populations where migration is allowed to happen in one direction, there are 29 states as shown in the following figure. The label for these two populations are 1 and 2, and migration only happens from population 2 to 1.

```    [ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 0 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [1] ], [ 1, [], [2] ]]: 1 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [1] ], [ 2, [], [2] ]]: 2 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 1, [], [2] ], [ 2, [2], [] ]]: 3 ],
[ [[ 1, [1], [] ], [ 1, [2], [1] ], [ 1, [], [2] ]]: 4 ],
[ [[ 1, [1], [2] ], [ 1, [2], [] ], [ 1, [], [1] ]]: 5 ],
[ [[ 1, [1], [] ], [ 1, [2], [2] ], [ 1, [], [1] ]]: 6 ],
[ [[ 1, [1], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 7 ],
[ [[ 1, [1], [2] ], [ 1, [], [1] ], [ 2, [2], [] ]]: 8 ],
[ [[ 1, [1], [1] ], [ 1, [2], [] ], [ 1, [], [2] ]]: 9 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [2] ]]: 10 ],
[ [[ 1, [1], [] ], [ 1, [2], [1] ], [ 2, [], [2] ]]: 11 ],
[ [[ 1, [1], [1] ], [ 1, [2], [] ], [ 2, [], [2] ]]: 12 ],
[ [[ 1, [1], [1] ], [ 1, [], [2] ], [ 2, [2], [] ]]: 13 ],
[ [[ 1, [1], [2] ], [ 1, [2], [1] ]]: 14 ],
[ [[ 1, [1], [1] ], [ 1, [2], [2] ]]: 15 ],
[ [[ 1, [1], [1] ], [ 2, [2], [2] ]]: 16 ],
[ [[ 1, [1,2], [] ], [ 1, [], [1] ], [ 1, [], [2] ]]: 17 ],
[ [[ 1, [1,2], [] ], [ 1, [], [1] ], [ 2, [], [2] ]]: 18 ],
[ [[ 1, [1,2], [1] ], [ 2, [], [2] ]]: 19 ],
[ [[ 1, [1,2], [1] ], [ 1, [], [2] ]]: 20 ],
[ [[ 1, [1,2], [2] ], [ 1, [], [1] ]]: 21 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [1,2] ]]: 22 ],
[ [[ 1, [1], [] ], [ 1, [], [1,2] ], [ 2, [2], [] ]]: 23 ],
[ [[ 1, [1], [1,2] ], [ 1, [2], [] ]]: 24 ],
[ [[ 1, [1], [] ], [ 1, [2], [1,2] ]]: 25 ],
[ [[ 1, [1], [1,2] ], [ 2, [2], [] ]]: 26 ],
[ [[ 1, [1,2], [] ], [ 1, [], [1,2] ]]: 27 ],
[ [[ 1, [1,2], [1,2] ]]: 28 ]
```

Copied below is the corresponding rate matrix, where R is the recombination rate, C is the coalescent rate. The row sum is zero so the diagonal values are the negated sum of all other values on each row.

```    	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28
0	-	0	m21	m21	0	0	0	c1	0	0	c2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	-	0	0	c1	c1	c1	0	0	c1	0	0	0	0	0	0	0	c1	0	0	0	0	c1	0	0	0	0	0	0
2	0	m21	-	0	0	0	0	0	0	0	0	c1	c1	0	0	0	0	0	c1	0	0	0	0	0	0	0	0	0	0
3	0	m21	0	-	0	0	0	0	c1	0	0	0	0	c1	0	0	0	0	0	0	0	0	0	c1	0	0	0	0	0
4	0	r	0	0	-	0	0	0	0	0	0	0	0	0	c1	0	0	0	0	0	c1	0	0	0	0	c1	0	0	0
5	0	r	0	0	0	-	0	0	0	0	0	0	0	0	c1	0	0	0	0	0	0	c1	0	0	c1	0	0	0	0
6	0	r	0	0	0	0	-	0	0	0	0	0	0	0	0	c1	0	0	0	0	0	c1	0	0	0	c1	0	0	0
7	r	0	0	0	0	0	0	-	0	0	0	0	m21	m21	0	0	c2	0	0	0	0	0	0	0	0	0	0	0	0
8	0	0	0	r	0	m21	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	c1	0	0
9	0	r	0	0	0	0	0	0	0	-	0	0	0	0	0	c1	0	0	0	0	c1	0	0	0	c1	0	0	0	0
10	r	0	0	0	0	0	m21	0	0	0	-	0	0	0	0	0	c1	0	0	0	0	0	0	0	0	0	0	0	0
11	0	0	r	0	m21	0	0	0	0	0	0	-	0	0	0	0	0	0	0	c1	0	0	0	0	0	0	0	0	0
12	0	0	r	0	0	0	0	0	0	m21	0	0	-	0	0	0	0	0	0	c1	0	0	0	0	0	0	0	0	0
13	0	0	0	r	0	0	0	0	0	m21	0	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	c1	0	0
14	0	0	0	0	r	r	0	0	0	0	0	0	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	c1
15	0	0	0	0	0	0	r	0	0	r	0	0	0	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	c1
16	0	0	0	0	0	0	0	r	0	0	r	0	0	0	0	m21	-	0	0	0	0	0	0	0	0	0	0	0	0
17	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-	0	0	c1	c1	0	0	0	0	0	c1	0
18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	m21	-	c1	0	0	0	0	0	0	0	0	0
19	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	-	m21	0	0	0	0	0	0	0	0
20	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	0	0	-	0	0	0	0	0	0	0	c1
21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	0	0	0	-	0	0	0	0	0	0	c1
22	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-	0	c1	c1	0	c1	0
23	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	m21	-	0	0	c1	0	0
24	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	0	-	0	0	0	c1
25	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	0	0	-	0	0	c1
26	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	m21	0	-	0	0
27	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-	c1
28	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	r	-
```

## Two Population Migration CTMC

For two samples in two populations where migration happens, there are 94 states as shown in the following figure. The label for these two populations are 1 and 2.

```    [ [[ 1, [1], [] ], [ 1, [], [1] ], [ 1, [], [2] ], [ 2, [2], [] ]]: 0 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [2] ], [ 2, [], [1] ]]: 1 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [1] ], [ 2, [], [2] ]]: 2 ],
[ [[ 1, [2], [] ], [ 1, [], [1] ], [ 1, [], [2] ], [ 2, [1], [] ]]: 3 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 2, [], [1] ], [ 2, [], [2] ]]: 4 ],
[ [[ 1, [1], [] ], [ 1, [], [2] ], [ 2, [2], [] ], [ 2, [], [1] ]]: 5 ],
[ [[ 1, [2], [] ], [ 2, [1], [] ], [ 2, [], [1] ], [ 2, [], [2] ]]: 6 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 7 ],
[ [[ 2, [1], [] ], [ 2, [2], [] ], [ 2, [], [1] ], [ 2, [], [2] ]]: 8 ],
[ [[ 1, [1], [] ], [ 2, [2], [] ], [ 2, [], [1] ], [ 2, [], [2] ]]: 9 ],
[ [[ 1, [], [1] ], [ 1, [], [2] ], [ 2, [1], [] ], [ 2, [2], [] ]]: 10 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [1] ], [ 1, [], [2] ]]: 11 ],
[ [[ 1, [], [2] ], [ 2, [1], [] ], [ 2, [2], [] ], [ 2, [], [1] ]]: 12 ],
[ [[ 1, [2], [] ], [ 1, [], [1] ], [ 2, [1], [] ], [ 2, [], [2] ]]: 13 ],
[ [[ 1, [], [1] ], [ 2, [1], [] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 14 ],
[ [[ 1, [2], [] ], [ 1, [], [2] ], [ 2, [1], [] ], [ 2, [], [1] ]]: 15 ],
[ [[ 1, [2], [1] ], [ 2, [1], [] ], [ 2, [], [2] ]]: 16 ],
[ [[ 1, [2], [] ], [ 1, [], [2] ], [ 2, [1], [1] ]]: 17 ],
[ [[ 1, [], [1] ], [ 2, [1], [2] ], [ 2, [2], [] ]]: 18 ],
[ [[ 1, [], [1] ], [ 2, [1], [] ], [ 2, [2], [2] ]]: 19 ],
[ [[ 1, [2], [] ], [ 1, [], [1] ], [ 2, [1], [2] ]]: 20 ],
[ [[ 1, [2], [2] ], [ 2, [1], [] ], [ 2, [], [1] ]]: 21 ],
[ [[ 2, [1], [] ], [ 2, [2], [2] ], [ 2, [], [1] ]]: 22 ],
[ [[ 1, [], [2] ], [ 2, [1], [1] ], [ 2, [2], [] ]]: 23 ],
[ [[ 1, [], [2] ], [ 2, [1], [] ], [ 2, [2], [1] ]]: 24 ],
[ [[ 1, [1], [1] ], [ 1, [2], [] ], [ 1, [], [2] ]]: 25 ],
[ [[ 2, [1], [] ], [ 2, [2], [1] ], [ 2, [], [2] ]]: 26 ],
[ [[ 1, [2], [] ], [ 2, [1], [2] ], [ 2, [], [1] ]]: 27 ],
[ [[ 2, [1], [2] ], [ 2, [2], [] ], [ 2, [], [1] ]]: 28 ],
[ [[ 1, [1], [] ], [ 1, [2], [2] ], [ 2, [], [1] ]]: 29 ],
[ [[ 1, [1], [2] ], [ 1, [2], [] ], [ 2, [], [1] ]]: 30 ],
[ [[ 2, [1], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 31 ],
[ [[ 1, [1], [] ], [ 2, [2], [1] ], [ 2, [], [2] ]]: 32 ],
[ [[ 1, [1], [] ], [ 2, [2], [2] ], [ 2, [], [1] ]]: 33 ],
[ [[ 1, [1], [2] ], [ 1, [2], [] ], [ 1, [], [1] ]]: 34 ],
[ [[ 1, [1], [] ], [ 1, [], [1] ], [ 2, [2], [2] ]]: 35 ],
[ [[ 1, [2], [] ], [ 2, [1], [1] ], [ 2, [], [2] ]]: 36 ],
[ [[ 1, [1], [] ], [ 1, [2], [1] ], [ 1, [], [2] ]]: 37 ],
[ [[ 1, [1], [] ], [ 1, [2], [2] ], [ 1, [], [1] ]]: 38 ],
[ [[ 1, [1], [] ], [ 1, [], [2] ], [ 2, [2], [1] ]]: 39 ],
[ [[ 1, [1], [2] ], [ 2, [2], [] ], [ 2, [], [1] ]]: 40 ],
[ [[ 1, [2], [1] ], [ 1, [], [2] ], [ 2, [1], [] ]]: 41 ],
[ [[ 1, [2], [2] ], [ 1, [], [1] ], [ 2, [1], [] ]]: 42 ],
[ [[ 1, [1], [] ], [ 1, [2], [1] ], [ 2, [], [2] ]]: 43 ],
[ [[ 1, [1], [1] ], [ 2, [2], [] ], [ 2, [], [2] ]]: 44 ],
[ [[ 1, [1], [1] ], [ 1, [], [2] ], [ 2, [2], [] ]]: 45 ],
[ [[ 1, [1], [2] ], [ 1, [], [1] ], [ 2, [2], [] ]]: 46 ],
[ [[ 1, [1], [1] ], [ 1, [2], [] ], [ 2, [], [2] ]]: 47 ],
[ [[ 1, [1], [2] ], [ 2, [2], [1] ]]: 48 ],
[ [[ 1, [1], [1] ], [ 2, [2], [2] ]]: 49 ],
[ [[ 1, [1], [2] ], [ 1, [2], [1] ]]: 50 ],
[ [[ 1, [1], [1] ], [ 1, [2], [2] ]]: 51 ],
[ [[ 1, [2], [2] ], [ 2, [1], [1] ]]: 52 ],
[ [[ 2, [1], [1] ], [ 2, [2], [2] ]]: 53 ],
[ [[ 2, [1], [2] ], [ 2, [2], [1] ]]: 54 ],
[ [[ 1, [2], [1] ], [ 2, [1], [2] ]]: 55 ],
[ [[ 1, [], [1] ], [ 2, [1,2], [] ], [ 2, [], [2] ]]: 56 ],
[ [[ 1, [], [1] ], [ 1, [], [2] ], [ 2, [1,2], [] ]]: 57 ],
[ [[ 1, [], [2] ], [ 2, [1,2], [] ], [ 2, [], [1] ]]: 58 ],
[ [[ 1, [1,2], [] ], [ 1, [], [2] ], [ 2, [], [1] ]]: 59 ],
[ [[ 1, [1,2], [] ], [ 1, [], [1] ], [ 1, [], [2] ]]: 60 ],
[ [[ 1, [1,2], [] ], [ 1, [], [1] ], [ 2, [], [2] ]]: 61 ],
[ [[ 1, [1,2], [] ], [ 2, [], [1] ], [ 2, [], [2] ]]: 62 ],
[ [[ 2, [1,2], [] ], [ 2, [], [1] ], [ 2, [], [2] ]]: 63 ],
[ [[ 1, [], [1] ], [ 2, [1,2], [2] ]]: 64 ],
[ [[ 1, [1,2], [1] ], [ 1, [], [2] ]]: 65 ],
[ [[ 2, [1,2], [1] ], [ 2, [], [2] ]]: 66 ],
[ [[ 1, [1,2], [2] ], [ 1, [], [1] ]]: 67 ],
[ [[ 1, [1,2], [1] ], [ 2, [], [2] ]]: 68 ],
[ [[ 1, [], [2] ], [ 2, [1,2], [1] ]]: 69 ],
[ [[ 1, [1,2], [2] ], [ 2, [], [1] ]]: 70 ],
[ [[ 2, [1,2], [2] ], [ 2, [], [1] ]]: 71 ],
[ [[ 1, [2], [] ], [ 2, [1], [] ], [ 2, [], [1,2] ]]: 72 ],
[ [[ 1, [1], [] ], [ 2, [2], [] ], [ 2, [], [1,2] ]]: 73 ],
[ [[ 2, [1], [] ], [ 2, [2], [] ], [ 2, [], [1,2] ]]: 74 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 1, [], [1,2] ]]: 75 ],
[ [[ 1, [1], [] ], [ 1, [2], [] ], [ 2, [], [1,2] ]]: 76 ],
[ [[ 1, [2], [] ], [ 1, [], [1,2] ], [ 2, [1], [] ]]: 77 ],
[ [[ 1, [1], [] ], [ 1, [], [1,2] ], [ 2, [2], [] ]]: 78 ],
[ [[ 1, [], [1,2] ], [ 2, [1], [] ], [ 2, [2], [] ]]: 79 ],
[ [[ 2, [1], [1,2] ], [ 2, [2], [] ]]: 80 ],
[ [[ 1, [2], [] ], [ 2, [1], [1,2] ]]: 81 ],
[ [[ 1, [1], [] ], [ 2, [2], [1,2] ]]: 82 ],
[ [[ 2, [1], [] ], [ 2, [2], [1,2] ]]: 83 ],
[ [[ 1, [2], [1,2] ], [ 2, [1], [] ]]: 84 ],
[ [[ 1, [1], [1,2] ], [ 1, [2], [] ]]: 85 ],
[ [[ 1, [1], [] ], [ 1, [2], [1,2] ]]: 86 ],
[ [[ 1, [1], [1,2] ], [ 2, [2], [] ]]: 87 ],
[ [[ 1, [], [1,2] ], [ 2, [1,2], [] ]]: 88 ],
[ [[ 1, [1,2], [] ], [ 1, [], [1,2] ]]: 89 ],
[ [[ 1, [1,2], [] ], [ 2, [], [1,2] ]]: 90 ],
[ [[ 2, [1,2], [] ], [ 2, [], [1,2] ]]: 91 ],
[ [[ 1, [1,2], [1,2] ]]: 92 ],
[ [[ 2, [1,2], [1,2] ]]: 93 ]
```

Copied below is the corresponding rate matrix, where R is the recombination rate, C is the coalescent rate, M12 is the migration rate from population 1 to 2, and M21 is the migration rate from population 2 to 1. The row sum is zero so the diagonal values are the negated sum of all other values on each row.

```    	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55	56	57	58	59	60	61	62	63	64	65	66	67	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83	84	85	86	87	88	89	90	91	92	93
0	-	0	M21	M12	0	0	0	0	0	0	0	M12	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	-	0	0	M21	M21	0	0	0	0	0	0	0	M21	M12	0	0	C2	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	M12	0	-	0	0	0	0	M12	0	M12	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	C1	0	0	0	0	0	0	0	0	0	0	0	C1	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	M21	0	0	-	0	0	M12	0	M12	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
4	0	M12	0	0	-	0	0	M21	M12	M21	0	0	0	0	0	0	0	0	0	0	0	C2	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
5	0	M12	0	0	0	-	M12	0	0	M21	M21	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	M21	0	M21	-	0	0	0	0	M21	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
7	0	0	M21	0	M12	0	0	-	0	0	0	0	0	M12	0	M12	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
8	0	0	0	M21	M21	0	0	0	-	0	0	0	0	0	M12	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
9	0	0	M21	M12	M12	M12	0	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
10	0	0	M21	0	0	M12	0	0	0	0	-	M12	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
11	M21	0	0	0	0	0	M12	0	0	0	M21	-	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
12	0	0	0	0	0	0	0	0	0	0	0	M21	-	M21	M12	M21	0	0	0	0	0	0	0	C2	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
13	0	M12	0	0	0	0	0	M21	0	0	M21	0	M12	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
14	0	M21	0	0	0	0	M21	0	M21	0	0	0	M21	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	C2	C2	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
15	M21	0	0	0	0	0	0	M21	M12	0	0	0	M12	0	0	-	0	0	C1	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
16	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	-	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
17	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M21	-	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0
18	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	-	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	M21	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
19	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-	0	M21	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0
20	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	-	0	0	M12	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
21	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	-	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
22	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-	0	0	0	0	0	0	0	M21	0	0	0	0	M12	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
23	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	M21	0	0	-	0	0	0	0	0	M21	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0
24	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	M12	M12	0	0	0	0	0	-	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
25	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	-	0	0	M21	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0
26	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	-	0	0	M21	0	M21	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
27	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	-	0	0	0	0	0	M21	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
28	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	M21	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
29	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	M12	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
30	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0
31	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	-	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0
32	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	0	0	0	0	0	0	M21	-	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
33	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	M12	0	0	0	0	0	-	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0
34	0	0	R	0	0	0	0	0	0	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	-	0	0	0	0	0	M12	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C1	0	0	0	0	0	0	0	0	0	0
35	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	M21	0	0	M21	0	0	0	0	0	0	0	0	0	-	0	0	0	0	0	0	0	0	M21	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0
36	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	M21	0	0	M21	0	0	0	0	0	M21	0	0	0	-	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	C2	0	0	0	0	0	0	0	0	0	0	0	0
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79	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	M21	0	0	0	0	M12	M12	-	0	0	0	0	0	C1	0	0	0	0	0	0	0	0
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83	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	M12	0	0	-	0	0	0	M12	0	0	0	0	0	C1
84	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	0	0	0	0	0	0	0	0	0	0	-	M12	M12	0	0	0	0	0	0	C1
85	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	R	0	M12	0	0	M21	-	0	0	0	0	0	0	0	0
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```

## References

1. M. Slatkin, J. L. Pollack, The concordance of gene trees and species trees at two linked loci, Genetics 172 (3) (2006) 1979-1984.

2. K. Simonsen, G. Churchill, A Markov chain model of coalescence with recombination, Theor Popul Biol 52 (1) (1997) 43-59.

3. T. Mailund, J. Y. Dutheil, A. Hobolth, G. Lunter, M. H. Schierup, Estimating divergence time and ancestral effective population size of Bornean and Sumatran orangutan subspecies using a coalescent hidden Markov model, PLoS Genet 7 (3) (2011) e1001319.

4. A. Hobolth, J. L. Jensen, Markovian approximation to the finite loci coalescent with recombination along multiple sequences, Theor Popul Biol.