cotpi 55 - Partitioning an integer and its double

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cotpi

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Jun 28, 2012, 4:48:00 PM6/28/12
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A partition of a positive integer n is a list of positive
integers, ordered from largest to smallest, such that the sum of
the integers in the sequence is n. Each integer in the list is
called a part.

What is the ratio of the number of possible partitions of a
positive integer n to the number of possible partitions of 2n
into n parts?

--
Originally posted at: http://cotpi.com/p/55/
Correct solutions will be archived at the URL mentioned above.

Solutions to 'Gambling with a die': http://cotpi.com/p/54/#solutions

Ilan Mayer

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Jun 28, 2012, 5:26:53 PM6/28/12
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SPOILER

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Each partition of 2n into n parts contains numbers in the range 1 to n
+1.

Subtracting 1 from each number and discarding the 0s thus results in a
partition of n.

Conversely, any partition of n contains 1 to n numbers. Adding 0s to
partitions with less than n numbers to bring up the total to n and
then adding 1 to each number results in a partition of 2n into n
parts.

Thus the ratio is 1.



Please reply to ilan dot mayer at hotmail dot com

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__/\\ //\__ Ilan Mayer
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/__ __\ Toronto, Canada
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henh...@gmail.com

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Sep 18, 2022, 5:09:16 PMSep 18
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e.g.
4 into 2 parts ==> (3,1) (2,2)


WRONG> 6 into 2 parts ==> (5,1) (4,2) (3,3) <True but Not relevant to the Prob.

6 into 3 parts ==> (4,1,1) (3,2,1) (2,2,2)

henh...@gmail.com

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Sep 19, 2022, 1:05:03 PMSep 19
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On Sunday, September 18, 2022 at 2:09:16 PM UTC-7, henh...@gmail.com wrote:
> On Thursday, June 28, 2012 at 1:48:00 PM UTC-7, cotpi wrote:
> > A partition of a positive integer n is a list of positive
> > integers, ordered from largest to smallest, such that the sum of
> > the integers in the sequence is n. Each integer in the list is
> > called a part.
> >
> > What is the ratio of the number of possible partitions of a
> > positive integer n to the number of possible partitions of 2n
> > into n parts?
> >
> > --
> > Originally posted at: http://cotpi.com/p/55/
> > Correct solutions will be archived at the URL mentioned above.
> >
> > Solutions to 'Gambling with a die': http://cotpi.com/p/54/#solutions
>
>

this is a NICE problem !

i'll check out his (her) other problems


> > number of possible partitions of 2n into n parts
> e.g.
> 4 into 2 parts ==> (3,1) (2,2)
>
>
> WRONG> 6 into 2 parts ==> (5,1) (4,2) (3,3) <True but Not relevant to the Prob.
>
> 6 into 3 parts ==> (4,1,1) (3,2,1) (2,2,2)


i'd love to see Other problems about Partitions !
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