CombinatorialCase110 - Revision history

WikiSysop at 21:42, 21 October 2017

2017-10-21T21:42:33Z

← Older revision		Revision as of 17:42, 21 October 2017
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	* <apll>M</apll> labeled balls (1), <apll>N</apll> labeled boxes (1), at most one ball per box (0)		* <apll>M</apll> labeled balls (1), <apll>N</apll> labeled boxes (1), at most one ball per box (0)
	* Sensitive to <apll>⎕IO</apll>		* Sensitive to <apll>⎕IO</apll>
			* Allows Lexicographic and Gray Code order
	* Counted result is an integer scalar		* Counted result is an integer scalar
	* Generated result is an integer matrix.		* Generated result is an integer matrix.
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	<pre>		<pre>
	110 1‼3 3		110 1‼3 ⍝ Permutations in unspecified order
			1 2 3
			1 3 2
			3 1 2
			3 2 1
			2 3 1
			2 1 3
			110 2‼3 ⍝ Permutations in Lexicographic order when L=R
	1 2 3		1 2 3
			1 3 2
	2 1 3		2 1 3
	2 3 1		2 3 1
			3 1 2
			3 2 1
			110 3‼3 ⍝ Permutations in Gray Code order when L=R
			1 2 3
	1 3 2		1 3 2
	3 1 2		3 1 2
	3 2 1		3 2 1
			2 3 1
			2 1 3
	⍝ Permutations of length M of N items		⍝ Permutations of length M of N items
	⍝ Labeled balls & boxes, ≤1 # Balls per Box		⍝ Labeled balls & boxes, ≤1 # Balls per Box
Line 97:		Line 112:
	110 1‼3 3		110 1‼3 3
	1 2 3		1 2 3
	~~2 1 3~~
	~~2 3 1~~
	1 3 2		1 3 2
	3 1 2		3 1 2
	3 2 1		3 2 1
			2 3 1
			2 1 3
	110 1‼2 3		110 1‼2 3
	1 2		1 2
	2 1		2 1
			2 3
			3 2
	1 3		1 3
	3 1		3 1
	2 3		110 2‼2 3 ⍝ Lexicographic order for Permutations with L≠R not implemented as yet
	3 2		NONCE ERROR
			110 2‼2 3
			∧
			110 3‼2 3 ⍝ Gray Code order for Permutations with L≠R not implemented as yet
			NONCE ERROR
			110 3‼2 3
			∧
	110 1‼1 3		110 1‼1 3
	1		1

WikiSysop at 22:20, 14 May 2017

2017-05-14T22:20:05Z

← Older revision		Revision as of 18:20, 14 May 2017
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	This case produces '''<apll>L</apll>-Permutations of <apll>R</apll> items''' (also called '''Partial Permutations''' or '''Sequences Without Repetition'''), where when <apll>L=R</apll> produces the familiar permutations <apll>!R</apll>. The length of each permutation returned is always <apll>L</apll>.		This case produces '''<apll>M</apll>-Permutations of <apll>N</apll> items''' (also called '''Partial Permutations''' or '''Sequences Without Repetition'''), where when <apll>M=N</apll> produces the familiar permutations <apll>!N</apll>. The length of each permutation returned is always <apll>M</apll>.

	* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at most one ball per box (0)		* <apll>M</apll> labeled balls (1), <apll>N</apll> labeled boxes (1), at most one ball per box (0)
	* Sensitive to <apll>⎕IO</apll>		* Sensitive to <apll>⎕IO</apll>
	* 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) R</apll> where <apll>!⍠(L) R</apll> calculates the [[Variant#Rising_and_Falling_Factorials\|'''Rising and Falling Factorial''']].		The count for this function is <apll>!⍠(-M) N</apll> where <apll>!⍠(M) N</apll> calculates the [[Variant#Rising_and_Falling_Factorials\|'''Rising and Falling Factorial''']].

	For example:		For example:
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	3 1 2		3 1 2
	3 2 1		3 2 1
	⍝ Permutations of length L of R items		⍝ Permutations of length M of N items
	⍝ Labeled balls & boxes, ≤1 # Balls per Box		⍝ Labeled balls & boxes, ≤1 # Balls per Box
	!3		!3
Line 115:		Line 115:
	</pre>		</pre>

	A function to calculate the permutations of <apll>R</apll> items could be defined as		A function to calculate the permutations of <apll>M</apll> items could be defined as

	<pre>		<pre>
	perm←{110 1‼⍵ ⍵}		perm←{110 1‼⍵}
	</pre>		</pre>

WikiSysop at 09:55, 30 April 2017

2017-04-30T09:55:56Z

← Older revision		Revision as of 05:55, 30 April 2017
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	* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at most one ball per box (0)		* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at most one ball per box (0)
	* Sensitive to ⎕IO		* Sensitive to <apll>⎕IO</apll>
	* 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))R</apll> where <apll>(~~!⍠L~~)R</apll> calculates the [[Variant#Rising_and_Falling_Factorials\|'''Rising and Falling Factorial''']].		The count for this function is <apll>!⍠(-L) R</apll> where <apll>!⍠(L) R</apll> calculates the [[Variant#Rising_and_Falling_Factorials\|'''Rising and Falling Factorial''']].

	For example:		For example:

WikiSysop at 01:47, 30 April 2017

2017-04-30T01:47:43Z

← Older revision		Revision as of 21:47, 29 April 2017
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	This case produces <apll>L</apll>-~~'''~~Permutations''' (also called '''Partial Permutations''' or '''Sequences Without Repetition''') ~~of <apll>R</apll> items~~, where when <apll>L=R</apll> produces the familiar permutations <apll>!R</apll>. The length of each permutation returned is always <apll>L</apll>.		This case produces '''<apll>L</apll>-Permutations of <apll>R</apll> items''' (also called '''Partial Permutations''' or '''Sequences Without Repetition'''), where when <apll>L=R</apll> produces the familiar permutations <apll>!R</apll>. The length of each permutation returned is always <apll>L</apll>.

	* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at most one ball per box (0)		* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at most one ball per box (0)

WikiSysop at 01:18, 30 April 2017

2017-04-30T01:18:33Z

← Older revision		Revision as of 21:18, 29 April 2017
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	3 2 1		3 2 1
	⍝ Permutations of length L of R items		⍝ Permutations of length L of R items
	⍝ Labeled balls & boxes, ≤1 #~~bpb~~		⍝ Labeled balls & boxes, ≤1 # Balls per Box
	!3		!3
	6		6

WikiSysop at 23:03, 29 April 2017

2017-04-29T23:03:53Z

← Older revision		Revision as of 19:03, 29 April 2017
Line 6:		Line 6:
	* Generated result is an integer matrix.		* Generated result is an integer matrix.

	The count for this function is <apll>(!⍠(-L))R</apll> where <apll>(!⍠L)R</apll> calculates the [[Variant\|~~rising or falling factorial~~]].		The count for this function is <apll>(!⍠(-L))R</apll> where <apll>(!⍠L)R</apll> calculates the [[Variant#Rising_and_Falling_Factorials\|'''Rising and Falling Factorial''']].

	For example:		For example:

WikiSysop: Created page with "This case produces L-'''Permutations''' (also called '''Partial Permutations''' or '''Sequences Without Repetition''') of R items, where when L..."

2017-04-29T22:56:56Z

Created page with "This case produces <apll>L</apll>-'''Permutations''' (also called '''Partial Permutations''' or '''Sequences Without Repetition''') of <apll>R</apll> items, where when <apll>L..."

New page

This case produces <apll>L</apll>-'''Permutations''' (also called '''Partial Permutations''' or '''Sequences Without Repetition''') of <apll>R</apll> items, where when <apll>L=R</apll> produces the familiar permutations <apll>!R</apll>. The length of each permutation returned is always <apll>L</apll>.

* <apll>L</apll> labeled balls (1), <apll>R</apll> labeled boxes (1), at most one ball per box (0)
* Sensitive to ⎕IO
* Counted result is an integer scalar
* Generated result is an integer matrix.

The count for this function is <apll>(!⍠(-L))R</apll> where <apll>(!⍠L)R</apll> calculates the [[Variant|rising or falling factorial]].

For example:

If we have 3 labeled balls (❶❷❸) and 3 labeled boxes (<apll>123</apll>) with at most one ball per box, there are <apll>6</apll> (<apll>↔ (!⍠¯3)3 ↔ 3×2×1</apll>) ways to meet these criteria:

{| border="0" cellpadding="5" cellspacing="0"
|
{| border="1" cellpadding="5" cellspacing="0"
|❶
|❷
|❸
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|❷
|❶
|❸
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|❷
|❸
|❶
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|❶
|❸
|❷
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|❸
|❶
|❷
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|
{| border="1" cellpadding="5" cellspacing="0"
|❸
|❷
|❶
|-
|align="center"|1
|align="center"|2
|align="center"|3
|}
|}

The diagram above corresponds to

<pre>
110 1‼3 3
1 2 3
2 1 3
2 3 1
1 3 2
3 1 2
3 2 1
⍝ Permutations of length L of R items
⍝ Labeled balls & boxes, ≤1 #bpb
!3
6
110‼3 3
6
110 0‼3 3
6
⍴110 1‼3 3
6 3
110 1‼3 3
1 2 3
2 1 3
2 3 1
1 3 2
3 1 2
3 2 1
110 1‼2 3
1 2
2 1
1 3
3 1
2 3
3 2
110 1‼1 3
1
2
3
</pre>

A function to calculate the permutations of <apll>R</apll> items could be defined as

<pre>
perm←{110 1‼⍵ ⍵}
</pre>