CombinatorialCase001 - Revision history

WikiSysop at 23:49, 19 June 2017

2017-06-19T23:49:33Z

← Older revision		Revision as of 19:49, 19 June 2017
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	* Generated result is a nested vector of integer vectors.		* Generated result is a nested vector of integer vectors.

	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.		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)#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 16:14, 14 May 2017

2017-05-14T16:14:13Z

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

	* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)		* <apll>M</apll> unlabeled balls (0), <apll>N</apll> unlabeled boxes (0), any # of balls per box (1)
	* 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 a nested vector of integer vectors.		* Generated result is a nested vector of integer vectors.

	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 exactly <apll>R</apll> parts.		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.

	For example:		For example:
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	3 2 1		3 2 1
	2 2 2		2 2 2
	⍝ Partitions of L into at most R parts		⍝ Partitions of M into at most N parts
	⍝ Unlabeled balls & boxes, any # Balls per Box		⍝ Unlabeled balls & boxes, any # Balls per Box
	⍪001 1‼5 5		⍪001 1‼5 5
Line 128:		Line 128:
	==Identities==		==Identities==

	As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.		As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(M+N)PN N ↔ +/M PN¨0..N</apll>.

	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:		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:

	<apll>001 ~~1‼L R~~ ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll>		<apll>001 1‼M N ↔ (⊂[⎕IO+1] ¯1+002 1‼(M+N) N)~¨0</apll>

WikiSysop at 01:43, 30 April 2017

2017-04-30T01:43:30Z

← Older revision		Revision as of 21:43, 29 April 2017
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	This case produces the '''Partitions of the number~~'''~~ L into at most R parts.		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)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)

WikiSysop at 01:15, 30 April 2017

2017-04-30T01:15:28Z

← Older revision		Revision as of 21:15, 29 April 2017
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	2 2 2		2 2 2
	⍝ Partitions of L into at most R parts		⍝ Partitions of L into at most R parts
	⍝ Unlabeled balls & boxes, any #~~bpb~~		⍝ Unlabeled balls & boxes, any # Balls per Box
	⍪001 1‼5 5		⍪001 1‼5 5
	5		5

WikiSysop at 22:37, 29 April 2017

2017-04-29T22:37:13Z

← Older revision		Revision as of 18:37, 29 April 2017
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	This case produces the Partitions of the number L into at most R parts.		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)		* <apll>L</apll> unlabeled balls (0), <apll>R</apll> unlabeled boxes (0), any # of balls per box (1)

WikiSysop: /* Identities */

2017-04-29T20:53:36Z

Identities

← Older revision		Revision as of 16:53, 29 April 2017
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	As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.		As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.

	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:		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:

	<apll>001 1‼L R ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll>		<apll>001 1‼L R ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll>

WikiSysop at 19:27, 29 April 2017

2017-04-29T19:27:37Z

← Older revision		Revision as of 15:27, 29 April 2017
Line 6:		Line 6:
	* Generated result is a nested vector of integer vectors.		* Generated result is a nested vector of integer vectors.

	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.		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 exactly <apll>R</apll> parts.

	For example:		For example:

WikiSysop: Created page with "This case produces the Partitions of the number L into at most R parts. * L unlabeled balls (0), R unlabeled boxes (0), any # of balls per box (1) *..."

2017-04-29T19:19:45Z

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) *..."

New page

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)
* Not <apll>⎕IO</apll>-sensitive
* Counted result is an integer scalar
* Generated result is a nested vector of integer vectors.

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.

For example:

If we have <apll>6</apll> unlabeled balls (●●●●●●) 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:

{| border="0" cellpadding="5" cellspacing="0"
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{| border="1" cellpadding="5" cellspacing="0"
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{| border="1" cellpadding="5" cellspacing="0"
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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 the nested array

<pre>
⍪001 1‼6 3
6
5 1
4 2
4 1 1
3 3
3 2 1
2 2 2
⍝ Partitions of L into at most R parts
⍝ Unlabeled balls & boxes, any #bpb
⍪001 1‼5 5
5
4 1
3 2
3 1 1
2 2 1
2 1 1 1
1 1 1 1 1
⍪001 1‼5 4
5
4 1
3 2
3 1 1
2 2 1
2 1 1 1
⍪001 1‼5 3
5
4 1
3 2
3 1 1
2 2 1
⍪001 1‼5 2
5
4 1
3 2
⍪001 1‼5 1
5</pre>

==Identities==

As shown in [https://en.wikipedia.org/wiki/Twelvefold_way#case_fnx Wikipedia], <apll>(L+R)PN R ↔ +/L PN¨0..R</apll>.

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:

<apll>001 1‼L R ↔ (⊂[⎕IO+1] ¯1+002 1‼(L+R) R)~¨0</apll>