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Revision as of 13:41, 31 January 2015
, 13:41, 31 January 2015→Derivation for getting the index of a genotype in an enumerated list
An important observation here is that for the enumeration of A alleles for a given P ploidy, the enumeration of A-1 alleles for P ploidy is a subsequence.
An important observation here is that for the enumeration of A alleles for a given P ploidy, the enumeration of A-1 alleles for P ploidy is a subsequence.
{| class="wikitable"
{| class="wikitable"
| AAA <br>
| AAA <br>
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
|-
|}
a_1, ... a_P is ordered and indexed 0 to A-1. Clearly when P = 1, the enumeration is of the genotypes is trivially the same as the allele.
when P > 1,
<math>
\begin{align}
G(a_1,.. , a_P) &= \| \{\text{genotypes of } a_P \text{ alleles for ploidy P}\} \| + G(a_1,.. , a_{P-1})\\
&= F(P, a_P) + G(a_1,..,a_{P-1})
\end{align}
</math>
This gives a recursive relationship that is a chain of P-1 calculations.
{| class="wikitable"
|-
! scope="col"| Index
! scope="col"| A=4,P=3
! scope="col"| A=3,P=3
! scope="col"| A=2,P=3
! scope="col"| A=1,P=3
! scope="col"| A=1,P=3
|-
|1 <br>
2 <br>
3 <br>
4 <br>
5 <br>
6 <br>
7 <br>
8 <br>
9 <br>
10 <br>
11 <br>
12 <br>
13 <br>
14 <br>
15 <br>
16 <br>
| AAA <br>
AAB <br>
ABB <br>
BBB <br>
AAC <br>
ABC <br>
BBC <br>
ACC <br>
BCC <br>
CCC <br>
AAD <br>
ABD <br>
BBD <br>
ACD <br>
BCD <br>
CCD
| AAA <br>
AAB <br>
ABB <br>
BBB <br>
AAC <br>
ABC <br>
BBC <br>
ACC <br>
BCC <br>
CCC <br><br><br><br><br><br><br>
| <br><br><br><br><br><br> <br><br><br><br>
AA <br>
AB <br>
BB <br>
AC <br>
BC <br>
CC <br>
| <br><br><br><br><br><br> <br><br><br><br>
AA <br>
AB <br>
BB <br>
<br><br><br>
| <br><br><br><br><br><br> <br><br><br><br><br><br><br>
A <br>
B <br>
C <br>
|-
|
|G(CCD)
|F(3,3)
|G(CC)
|F(2,2)
|G(C)
|-
|
|
|10
|
|3
|3
|-
|-
|}
|}
