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| | | |
| 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. | | 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. |
| + | |
| + | The nested genotype sequence is in red. The blue sequence shows the sequence of genotypes enumerated without involving <math>a_P</math>. |
| | | |
| {| class="wikitable" | | {| class="wikitable" |
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| ! scope="col"| Index | | ! scope="col"| Index |
| ! scope="col"| A=4,P=3 | | ! scope="col"| A=4,P=3 |
− | ! scope="col"| A=3,P=3 | + | ! scope="col"| A=4,P=2 |
− | ! scope="col"| A=2,P=3 | + | ! scope="col"| A=4,P=1 |
− | ! scope="col"| A=1,P=3
| |
| |- | | |- |
| |0 <br> | | |0 <br> |
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| 18 <br> | | 18 <br> |
| 19 <br> | | 19 <br> |
− | | AAA<br> | + | | <span style="color:#FF0000">AAA</span><br> |
− | AAB <br> | + | <span style="color:#FF0000">AAB</span> <br> |
− | ABB <br> | + | <span style="color:#FF0000">ABB</span> <br> |
− | BBB <br> | + | <span style="color:#FF0000">BBB</span> <br> |
− | AAC <br> | + | <span style="color:#FF0000">AAC</span> <br> |
− | ABC <br> | + | <span style="color:#FF0000">ABC</span> <br> |
− | BBC <br> | + | <span style="color:#FF0000">BBC</span> <br> |
− | ACC <br> | + | <span style="color:#FF0000">ACC</span> <br> |
− | BCC <br> | + | <span style="color:#FF0000">BCC</span> <br> |
− | CCC <br> | + | <span style="color:#FF0000">CCC</span> <br> |
| + | <span style="color:#FF0000">AAD</span> <br> |
| + | <span style="color:#FF0000">ABD</span> <br> |
| + | <span style="color:#FF0000">BBD</span> <br> |
| + | <span style="color:#FF0000">ACD</span> <br> |
| + | <span style="color:#FF0000">BCD</span> <br> |
| + | <span style="color:#FF0000">CCD</span> <br> |
| + | <span style="color:#FF0000">ADD</span> <br> |
| + | <span style="color:#FF0000">BDD</span> <br> |
| + | <span style="color:#FF0000">CDD</span> <br> |
| + | <span style="color:#FF0000">DDD</span> <br> |
| + | |<span style="color:#0000FF">AAA</span><br> |
| + | <span style="color:#0000FF">AAB</span> <br> |
| + | <span style="color:#0000FF">ABB</span> <br> |
| + | <span style="color:#0000FF">BBB</span> <br> |
| + | <span style="color:#0000FF">AAC</span> <br> |
| + | <span style="color:#0000FF">ABC</span> <br> |
| + | <span style="color:#0000FF">BBC</span> <br> |
| + | <span style="color:#0000FF">ACC</span> <br> |
| + | <span style="color:#0000FF">BCC</span> <br> |
| + | <span style="color:#0000FF">CCC</span> <br> |
| <span style="color:#FF0000">AA</span>D <br> | | <span style="color:#FF0000">AA</span>D <br> |
| <span style="color:#FF0000">AB</span>D <br> | | <span style="color:#FF0000">AB</span>D <br> |
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| BCC <br> | | BCC <br> |
| CCC <br> | | CCC <br> |
− | <span style="color:#FF0000">AA</span>D <br> | + | <span style="color:#0000FF">AA</span>D <br> |
− | <span style="color:#FF0000">AB</span>D <br> | + | <span style="color:#0000FF">AB</span>D <br> |
− | <span style="color:#FF0000">BB</span>D <br> | + | <span style="color:#0000FF">BB</span>D <br> |
− | <span style="color:#FF0000">AC</span>D <br> | + | <span style="color:#0000FF">AC</span>D <br> |
− | <span style="color:#FF0000">BC</span>D <br> | + | <span style="color:#0000FF">BC</span>D <br> |
− | <span style="color:#FF0000">CC</span>D <br> | + | <span style="color:#0000FF">CC</span>D <br> |
− | <span style="color:#FF0000">AD</span>D <br> | + | <span style="color:#FF0000">A</span>DD <br> |
− | <span style="color:#FF0000">BD</span>D <br> | + | <span style="color:#FF0000">B</span>DD <br> |
− | <span style="color:#FF0000">CD</span>D <br> | + | <span style="color:#FF0000">C</span>DD <br> |
− | <span style="color:#FF0000">DD</span>D <br> | + | <span style="color:#FF0000">D</span>DD <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 == | | == Derivation == |
| | | |