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 Post subject: Generating Function QuestionsPosted: Sat, 4 Aug 2012 02:44:44 UTC
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Joined: Wed, 4 Apr 2012 03:51:40 UTC
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Location: Hockeytown aka Detroit
Hmmm I guess generating functions aren't my thing~ I may have to post more questions as the weekend goes by :O

Question #1:
Show that the number of compositions of n into an odd number of parts, each of which is at least , is [].

My Work:
Well I didn't know what a composition of a positive integer was so I had to look it up. Then I found out that it is the number of ordered partitions of . I thought about subtracting from each part, but I don't see how that's going to help with the fact that there are an odd number of parts... I tried summing based on the number of odd parts, but that wasn't working for me because the values the parts can take change based on the number of parts.

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 Post subject: Re: Generating Function QuestionsPosted: Sat, 4 Aug 2012 10:51:03 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
Hmmm I guess generating functions aren't my thing~ I may have to post more questions as the weekend goes by :O

Question #1:
Show that the number of compositions of n into an odd number of parts, each of which is at least , is [].

My Work:
Well I didn't know what a composition of a positive integer was so I had to look it up. Then I found out that it is the number of ordered partitions of . I thought about subtracting from each part, but I don't see how that's going to help with the fact that there are an odd number of parts... I tried summing based on the number of odd parts, but that wasn't working for me because the values the parts can take change based on the number of parts.

I don't believe it is true --- which cannot be the generating function: there is only 2 ways to partition 10 into odd number of parts each at least 3, viz. (10) and (4,3,3).

What you want is obviously . I'm not sure this has a nicer expression, indeed OEIS sequence A027195 doesn't give anything.

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 Post subject: Re: Generating Function QuestionsPosted: Sat, 4 Aug 2012 17:50:29 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
Hmmm I see how it works a little more now. Does mark anything in the generating function (e.g. when considering the number of partitions of with parts, we let mark a part)?

outermeasure wrote:
2 ways to partition 10 into odd number of parts each at least 3, viz. (10) and (4,3,3).

Sorry, I should have explained what a composition was. A composition (or an ordered partion) is different from a partition.

Hence, and the number of ways is indeed .

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math puns are the first sine of madness
-JDR

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 Post subject: Re: Generating Function QuestionsPosted: Sun, 5 Aug 2012 04:55:24 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
Hmmm I see how it works a little more now. Does mark anything in the generating function (e.g. when considering the number of partitions of with parts, we let mark a part)?

outermeasure wrote:
2 ways to partition 10 into odd number of parts each at least 3, viz. (10) and (4,3,3).

Sorry, I should have explained what a composition was. A composition (or an ordered partion) is different from a partition.

Hence, and the number of ways is indeed .

Oh, ordered partitions... so just count the combinations using the usual trick k-1 (111|)'s, 1 final (111) (which doesn't count) and however many 1's and sum them all. Shouldn't be a problem.

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