CombinatorialCase002 - Revision history

WikiSysop at 23:50, 19 June 2017

2017-06-19T23:50:14Z

← Older revision		Revision as of 19:50, 19 June 2017
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	* Generated result is an integer matrix.		* Generated result is an integer matrix.

	The count for this function is <apll>M 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.		The count for this function is <apll>M PN N</apll> where <apll>M PN N</apll> calculates the number of [https://en.wikipedia.org/wiki/Partition_(number_theory)#Restricted_part_size_or_number_of_parts Partitions] of the number <apll>M</apll> into exactly <apll>N</apll> parts.

	For example:		For example:

WikiSysop at 16:17, 14 May 2017

2017-05-14T16:17:06Z

← Older revision		Revision as of 12:17, 14 May 2017
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	This case produces the '''Partitions of the number L into exactly R parts'''.		This case produces the '''Partitions of the number M into exactly N parts'''.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> unlabeled boxes (0), at least one ball per box (2)
	* Not <apll>⎕IO</apll>-sensitive		* Not <apll>⎕IO</apll>-sensitive
	* Counted result is an integer scalar		* Counted result is an integer scalar
	* Generated result is an integer matrix.		* Generated result is an integer matrix.

	The count for this function is <apll>L 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 exactly <apll>R</apll> parts.		The count for this function is <apll>M 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.

	For example:		For example:
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	3 3 2		3 3 2

	⍝ Partitions of L into R parts		⍝ Partitions of M into N parts
	⍝ Unlabeled balls & boxes, ≥1 # Balls per Box		⍝ Unlabeled balls & boxes, ≥1 # Balls per Box
	002 1‼5 5		002 1‼5 5
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	==Identities==		==Identities==

	Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts ([[CombinatorialCase001\|<apll>001</apll>]]) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts ([[CombinatorialCase002\|<apll>002</apll>]]), these cases are related by the following identity (after sorting the rows):		Because partitions of <apll>M</apll> into <apll>N</apll> non-negative parts ([[CombinatorialCase001\|<apll>001</apll>]]) is the same as partitions of <apll>M+N</apll> into <apll>N</apll> positive parts ([[CombinatorialCase002\|<apll>002</apll>]]), these cases are related by the following identity (after sorting the rows):

	<apll>002 ~~1‼L R~~ ↔ ⊃1+R↑¨001 1‼(~~0⌈L~~-R) R</apll>		<apll>002 1‼M N ↔ ⊃1+R↑¨001 1‼(0⌈M-N) N</apll>

WikiSysop at 01:43, 30 April 2017

2017-04-30T01:43:54Z

← Older revision		Revision as of 21:43, 29 April 2017
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	This case produces the '''Partitions of the number~~'''~~ L into exactly R parts.		This case produces the '''Partitions of the number L into exactly R parts'''.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)

WikiSysop at 01:16, 30 April 2017

2017-04-30T01:16:06Z

← Older revision		Revision as of 21:16, 29 April 2017
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	⍝ Partitions of L into R parts		⍝ Partitions of L into R parts
	⍝ Unlabeled balls & boxes, ≥1 #~~bpb~~		⍝ Unlabeled balls & boxes, ≥1 # Balls per Box
	002 1‼5 5		002 1‼5 5
	1 1 1 1 1		1 1 1 1 1

WikiSysop at 22:38, 29 April 2017

2017-04-29T22:38:14Z

← Older revision		Revision as of 18:38, 29 April 2017
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	This case produces the Partitions of the number L into exactly R parts.		This case produces the '''Partitions of the number''' L into exactly R parts.

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)

WikiSysop: /* Identities */

2017-04-29T20:56:06Z

Identities

← Older revision		Revision as of 16:56, 29 April 2017
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	==Identities==		==Identities==

	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 (after sorting the rows):		Because partitions of <apll>L</apll> into <apll>R</apll> non-negative parts ([[CombinatorialCase001\|<apll>001</apll>]]) is the same as partitions of <apll>L+R</apll> into <apll>R</apll> positive parts ([[CombinatorialCase002\|<apll>002</apll>]]), these cases are related by the following identity (after sorting the rows):

	<apll>002 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R</apll>		<apll>002 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R</apll>

WikiSysop: Created page with "This case produces the Partitions of the number L into exactly R parts. * L unlabeled balls (0), R unlabeled boxes (0), at least one ball per box (2..."

2017-04-29T19:43:18Z

Created page with "This case produces the Partitions of the number L into exactly R parts. * <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2..."

New page

This case produces the Partitions of the number L into exactly R parts.

* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), at least one ball per box (2)
* Not <apll>⎕IO</apll>-sensitive
* Counted result is an integer scalar
* Generated result is an integer matrix.

The count for this function is <apll>L 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 exactly <apll>R</apll> parts.

For example:

If we have <apll>8</apll> unlabeled balls (●●●●●●●●) and <apll>3</apll> unlabeled boxes with at least one ball per box, there are <apll>5</apll> (<apll>↔ 8 PN 3</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
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{| border="1" cellpadding="5" cellspacing="0"
|● ● ● ● ● ●
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|}
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{| border="1" cellpadding="5" cellspacing="0"
|  ● ● ● ● ●
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|}
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{| border="1" cellpadding="5" cellspacing="0"
|    ● ● ● ●
|      ● ● ●
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|-
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|}
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{| border="1" cellpadding="5" cellspacing="0"
|    ● ● ● ●
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{| border="1" cellpadding="5" cellspacing="0"
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The diagram above corresponds to

<pre>
002 1‼8 3
6 1 1
5 2 1
4 3 1
4 2 2
3 3 2

⍝ Partitions of L into R parts
⍝ Unlabeled balls & boxes, ≥1 #bpb
002 1‼5 5
1 1 1 1 1
002 1‼5 4
2 1 1 1
002 1‼5 3
3 1 1
2 2 1
002 1‼5 2
4 1
3 2
002 1‼5 1
5</pre>

==Identities==

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 (after sorting the rows):

<apll>002 1‼L R ↔ ⊃1+R↑¨001 1‼(0⌈L-R) R</apll>