http://wiki.nars2000.org/index.php?title=CombinatorialCase001&feed=atom&action=history
CombinatorialCase001 - Revision history
2024-03-28T09:42:52Z
Revision history for this page on the wiki
MediaWiki 1.38.4
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3093&oldid=prev
WikiSysop at 23:49, 19 June 2017
2017-06-19T23:49:33Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:49, 19 June 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">Line 6:</td>
<td colspan="2" class="diff-lineno">Line 6:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Generated result is a nested vector of integer vectors.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Generated result is a nested vector of integer vectors.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The count for this function is <apll>(M+N)PN N</apll> where <apll>M PN N</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll>M</apll> into exactly <apll>N</apll> parts.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The count for this function is <apll>(M+N)PN N</apll> where <apll>M PN N</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory)<ins style="font-weight: bold; text-decoration: none;">#Restricted_part_size_or_number_of_parts </ins>Partitions] of the number <apll>M</apll> into exactly <apll>N</apll> parts.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example:</div></td></tr>
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WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3065&oldid=prev
WikiSysop at 16:14, 14 May 2017
2017-05-14T16:14:13Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:14, 14 May 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This case produces the '''Partitions of the number <del style="font-weight: bold; text-decoration: none;">L </del>into at most <del style="font-weight: bold; text-decoration: none;">R </del>parts'''.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This case produces the '''Partitions of the number <ins style="font-weight: bold; text-decoration: none;">M </ins>into at most <ins style="font-weight: bold; text-decoration: none;">N </ins>parts'''.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <apll><del style="font-weight: bold; text-decoration: none;">L</del></apll> unlabeled balls (0), <apll><del style="font-weight: bold; text-decoration: none;">R</del></apll> unlabeled boxes (0), any # of balls per box (1)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <apll><ins style="font-weight: bold; text-decoration: none;">M</ins></apll> unlabeled balls (0), <apll><ins style="font-weight: bold; text-decoration: none;">N</ins></apll> unlabeled boxes (0), any # of balls per box (1)</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Not <apll>⎕IO</apll>-sensitive</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Not <apll>⎕IO</apll>-sensitive</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Counted result is an integer scalar</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Counted result is an integer scalar</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Generated result is a nested vector of integer vectors.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Generated result is a nested vector of integer vectors.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The count for this function is <apll>(<del style="font-weight: bold; text-decoration: none;">L</del>+<del style="font-weight: bold; text-decoration: none;">R</del>)PN <del style="font-weight: bold; text-decoration: none;">R</del></apll> where <apll><del style="font-weight: bold; text-decoration: none;">L </del>PN <del style="font-weight: bold; text-decoration: none;">R</del></apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll><del style="font-weight: bold; text-decoration: none;">L</del></apll> into exactly <apll><del style="font-weight: bold; text-decoration: none;">R</del></apll> parts.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The count for this function is <apll>(<ins style="font-weight: bold; text-decoration: none;">M</ins>+<ins style="font-weight: bold; text-decoration: none;">N</ins>)PN <ins style="font-weight: bold; text-decoration: none;">N</ins></apll> where <apll><ins style="font-weight: bold; text-decoration: none;">M </ins>PN <ins style="font-weight: bold; text-decoration: none;">N</ins></apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll><ins style="font-weight: bold; text-decoration: none;">M</ins></apll> into exactly <apll><ins style="font-weight: bold; text-decoration: none;">N</ins></apll> parts.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l96">Line 96:</td>
<td colspan="2" class="diff-lineno">Line 96:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 3 2 1</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 3 2 1</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 2 2</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 2 2</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Partitions of <del style="font-weight: bold; text-decoration: none;">L </del>into at most <del style="font-weight: bold; text-decoration: none;">R </del>parts</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Partitions of <ins style="font-weight: bold; text-decoration: none;">M </ins>into at most <ins style="font-weight: bold; text-decoration: none;">N </ins>parts</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Unlabeled balls & boxes, any # Balls per Box</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Unlabeled balls & boxes, any # Balls per Box</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍪001 1‼5 5</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍪001 1‼5 5</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l128">Line 128:</td>
<td colspan="2" class="diff-lineno">Line 128:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Identities==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(<del style="font-weight: bold; text-decoration: none;">L</del>+<del style="font-weight: bold; text-decoration: none;">R</del>)PN <del style="font-weight: bold; text-decoration: none;">R </del>↔ +/<del style="font-weight: bold; text-decoration: none;">L </del>PN¨0..<del style="font-weight: bold; text-decoration: none;">R</del></apll>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(<ins style="font-weight: bold; text-decoration: none;">M</ins>+<ins style="font-weight: bold; text-decoration: none;">N</ins>)PN <ins style="font-weight: bold; text-decoration: none;">N </ins>↔ +/<ins style="font-weight: bold; text-decoration: none;">M </ins>PN¨0..<ins style="font-weight: bold; text-decoration: none;">N</ins></apll>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Because partitions of <apll><del style="font-weight: bold; text-decoration: none;">L</del></apll> into <apll><del style="font-weight: bold; text-decoration: none;">R</del></apll> non-negative parts ([[CombinatorialCase001|<apll>001</apll>]]) is the same as partitions of <apll><del style="font-weight: bold; text-decoration: none;">L</del>+<del style="font-weight: bold; text-decoration: none;">R</del></apll> into <apll><del style="font-weight: bold; text-decoration: none;">R</del></apll> positive parts ([[CombinatorialCase002|<apll>002</apll>]]), these cases are related by the following identity:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Because partitions of <apll><ins style="font-weight: bold; text-decoration: none;">M</ins></apll> into <apll><ins style="font-weight: bold; text-decoration: none;">N</ins></apll> non-negative parts ([[CombinatorialCase001|<apll>001</apll>]]) is the same as partitions of <apll><ins style="font-weight: bold; text-decoration: none;">M</ins>+<ins style="font-weight: bold; text-decoration: none;">N</ins></apll> into <apll><ins style="font-weight: bold; text-decoration: none;">N</ins></apll> positive parts ([[CombinatorialCase002|<apll>002</apll>]]), these cases are related by the following identity:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><apll>001 <del style="font-weight: bold; text-decoration: none;">1‼L R </del>↔ (⊂[⎕IO+1] ¯1+002 1‼(<del style="font-weight: bold; text-decoration: none;">L</del>+<del style="font-weight: bold; text-decoration: none;">R</del>) <del style="font-weight: bold; text-decoration: none;">R</del>)~¨0</apll></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><apll>001 <ins style="font-weight: bold; text-decoration: none;">1‼M N </ins>↔ (⊂[⎕IO+1] ¯1+002 1‼(<ins style="font-weight: bold; text-decoration: none;">M</ins>+<ins style="font-weight: bold; text-decoration: none;">N</ins>) <ins style="font-weight: bold; text-decoration: none;">N</ins>)~¨0</apll></div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3038&oldid=prev
WikiSysop at 01:43, 30 April 2017
2017-04-30T01:43:30Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:43, 30 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This case produces the '''Partitions of the number<del style="font-weight: bold; text-decoration: none;">''' </del>L into at most R parts.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This case produces the '''Partitions of the number L into at most R parts<ins style="font-weight: bold; text-decoration: none;">'''</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)</div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3028&oldid=prev
WikiSysop at 01:15, 30 April 2017
2017-04-30T01:15:28Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 01:15, 30 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l97">Line 97:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 2 2</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 2 2</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Partitions of L into at most R parts</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Partitions of L into at most R parts</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Unlabeled balls & boxes, any #<del style="font-weight: bold; text-decoration: none;">bpb</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> ⍝ Unlabeled balls & boxes, any # <ins style="font-weight: bold; text-decoration: none;">Balls per Box</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍪001 1‼5 5</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> ⍪001 1‼5 5</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 5</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 5</div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3018&oldid=prev
WikiSysop at 22:37, 29 April 2017
2017-04-29T22:37:13Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:37, 29 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This case produces the Partitions of the number L into at most R parts.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This case produces the <ins style="font-weight: bold; text-decoration: none;">'''</ins>Partitions of the number<ins style="font-weight: bold; text-decoration: none;">''' </ins>L into at most R parts.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)</div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3011&oldid=prev
WikiSysop: /* Identities */
2017-04-29T20:53:36Z
<p><span dir="auto"><span class="autocomment">Identities</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:53, 29 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l130">Line 130:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts (<apll>001</apll>) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts (<apll>002</apll>), these cases are related by the following identity:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts (<ins style="font-weight: bold; text-decoration: none;">[[CombinatorialCase001|</ins><apll>001</apll><ins style="font-weight: bold; text-decoration: none;">]]</ins>) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts (<ins style="font-weight: bold; text-decoration: none;">[[CombinatorialCase002|</ins><apll>002</apll><ins style="font-weight: bold; text-decoration: none;">]]</ins>), these cases are related by the following identity:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><apll>001 1‼L R ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><apll>001 1‼L R ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll></div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3008&oldid=prev
WikiSysop at 19:27, 29 April 2017
2017-04-29T19:27:37Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:27, 29 April 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">Line 6:</td>
<td colspan="2" class="diff-lineno">Line 6:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Generated result is a nested vector of integer vectors.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Generated result is a nested vector of integer vectors.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The count for this function is <apll>(L+R)PN R</apll> where <apll>L PN R</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll>L</apll> into <del style="font-weight: bold; text-decoration: none;">at most </del><apll>R</apll> parts.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The count for this function is <apll>(L+R)PN R</apll> where <apll>L PN R</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll>L</apll> into <ins style="font-weight: bold; text-decoration: none;">exactly </ins><apll>R</apll> parts.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For example:</div></td></tr>
</table>
WikiSysop
http://wiki.nars2000.org/index.php?title=CombinatorialCase001&diff=3007&oldid=prev
WikiSysop: Created page with "This case produces the Partitions of the number L into at most R parts. * <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1) *..."
2017-04-29T19:19:45Z
<p>Created page with "This case produces the Partitions of the number L into at most R parts. * <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1) *..."</p>
<p><b>New page</b></p><div>This case produces the Partitions of the number L into at most R parts.<br />
<br />
* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)<br />
* Not <apll>⎕IO</apll>-sensitive<br />
* Counted result is an integer scalar<br />
* Generated result is a nested vector of integer vectors.<br />
<br />
The count for this function is <apll>(L+R)PN R</apll> where <apll>L PN R</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory) Partitions] of the number <apll>L</apll> into at most <apll>R</apll> parts.<br />
<br />
For example:<br />
<br />
If we have <apll>6</apll> unlabeled balls (<span style="font-size: 2em;">●●●●●●</span>) and <apll>3</apll> unlabeled boxes with any # of balls per box, there are <apll>7</apll> (<apll>↔ (6+3)PN 3</apll>) ways to meet these criteria:<br />
<br />
{| border="0" cellpadding="5" cellspacing="0"<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|&nbsp;&nbsp;&nbsp;<br />
|&nbsp;&nbsp;&nbsp;<br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br />
|&nbsp;&nbsp;&nbsp;<br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|&nbsp;&nbsp;&nbsp;<br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|&nbsp;&nbsp;&nbsp;<br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|<br />
{| border="1" cellpadding="5" cellspacing="0"<br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|<span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">&nbsp;</span><br /><span style="font-size: 2em;">●</span><br /><span style="font-size: 2em;">●</span><br />
|-<br />
|&nbsp;<br />
|&nbsp;<br />
|&nbsp;<br />
|}<br />
|}<br />
<br />
The diagram above corresponds to the nested array<br />
<br />
<pre><br />
⍪001 1‼6 3<br />
6<br />
5 1<br />
4 2<br />
4 1 1<br />
3 3<br />
3 2 1<br />
2 2 2<br />
⍝ Partitions of L into at most R parts<br />
⍝ Unlabeled balls & boxes, any #bpb<br />
⍪001 1‼5 5<br />
5<br />
4 1<br />
3 2<br />
3 1 1<br />
2 2 1<br />
2 1 1 1<br />
1 1 1 1 1<br />
⍪001 1‼5 4<br />
5<br />
4 1<br />
3 2<br />
3 1 1<br />
2 2 1<br />
2 1 1 1<br />
⍪001 1‼5 3<br />
5<br />
4 1<br />
3 2<br />
3 1 1<br />
2 2 1<br />
⍪001 1‼5 2<br />
5<br />
4 1<br />
3 2<br />
⍪001 1‼5 1<br />
5</pre><br />
<br />
==Identities==<br />
<br />
As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.<br />
<br />
Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts (<apll>001</apll>) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts (<apll>002</apll>), these cases are related by the following identity:<br />
<br />
<apll>001 1‼L R ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll></div>
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