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Revision as of 13:33, 2 February 2015
, 13:33, 2 February 2015→Derivation for getting the index of a genotype in an enumerated list
= Derivation for getting the index of a genotype in an enumerated list =
= Derivation for getting the index of a genotype in an enumerated list =
== Observation of nested patterns ==
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.
! scope="col"| A=1,P=3
! scope="col"| A=1,P=3
|-
|-
|1 <br>
|0 <br>
1 <br>
2 <br>
2 <br>
3 <br>
3 <br>
18 <br>
18 <br>
19 <br>
19 <br>
| AAA <br>
| AAA <br>
AAB <br>
AAB <br>
|}
|}
Another important observation here is that for a genotype <math>(a_1, a_2, .. a_P)</math>, The <math>F(P, a_p)</math>th genotype to <math>(a_1, a_2, .. a_P)</math> all end with <math>a_p</math>, this does not help distinguish the order, so we need to only examine the genotype <math>(a_1,..., a_{P-1})</math>.
Another important observation here is that for a genotype <math>(a_1, a_2, .. a_P)</math>, The <math>F(P, a_p)</math>th genotype to <math>(a_1, a_2, .. a_P)</math> all end with <math>a_p</math>, this does not help distinguish the order, so we need to only examine the genotype <math>(a_1,..., a_{P-1})</math>. The sub genotype <math>(a_1,..., a_{P-1})</math> is also ordered else <math>(a_1,..., a_{P})</math> is not ordered.
{| 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
|-
|0 <br>
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>
17 <br>
18 <br>
19 <br>
| AAA<br>
AAB <br>
ABB <br>
BBB <br>
AAC <br>
ABC <br>
BBC <br>
ACC <br>
BCC <br>
CCC <br>
<span style="color:#FF0000">AA</span>D <br>
<span style="color:#FF0000">AB</span>D <br>
<span style="color:#FF0000">BB</span>D <br>
<span style="color:#FF0000">AC</span>D <br>
<span style="color:#FF0000">BC</span>D <br>
<span style="color:#FF0000">CC</span>D <br>
<span style="color:#FF0000">AD</span>D <br>
<span style="color:#FF0000">BD</span>D <br>
<span style="color:#FF0000">CD</span>D <br>
<span style="color:#FF0000">DD</span>D <br>
| AAA<br>
AAB <br>
ABB <br>
BBB <br>
AAC <br>
ABC <br>
BBC <br>
ACC <br>
BCC <br>
CCC <br>
<span style="color:#FF0000">AA</span>D <br>
<span style="color:#FF0000">AB</span>D <br>
<span style="color:#FF0000">BB</span>D <br>
<span style="color:#FF0000">AC</span>D <br>
<span style="color:#FF0000">BC</span>D <br>
<span style="color:#FF0000">CC</span>D <br>
<span style="color:#FF0000">AD</span>D <br>
<span style="color:#FF0000">BD</span>D <br>
<span style="color:#FF0000">CD</span>D <br>
<span style="color:#FF0000">DD</span>D <br>
| AAA <br>
AAB <br>
ABB <br>
BBB <br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
| AAA <br>
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>
|-
|}
== Derivation ==
<math>a_1, ... a_P</math> is ordered and indexed 0 to A-1. Clearly when P = 1, the enumeration of the genotypes is trivially the same as the allele.
<math>a_1, ... a_P</math> is ordered and indexed 0 to A-1. Clearly when P = 1, the enumeration of the genotypes is trivially the same as the allele.
