In continuation of my previous post on perfect partitions, I seek to extend the previous result to all numbers, not just one less than a prime power.
Previously, we had:
The number of perfect partitions of the number
is the same as the number of compositions of the number .
Today, we will find a relation for the number of perfect partitions of any
positive integer
Parallel to the last result, we can see that the number of perfect partitions
of the number
This can immediately be proved by extending the previous post: still use finite geometric series and the positivity of the terms to show that each exponent can be reached in exactly one way. The different factors would again be of the form:
In this fashion, we see that the number of perfect partitions are the same as
the number of ordered factorizations of
As an aside that made more sense in the original post, I consider the number of compositions of a number. How many compositions are there for a number n?
I hadn't seen this before, but upon consideration I see that it is a very
simple exercise that one might encounter. Imagine we are thinking of the number
of compositions of n. Then
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Comments (2)
2011-04-26 Abhay
If I understood your post correctly. Suppose I wish to find perfect partitions of 95, so i can write:
95 = [ (2^5) * 3 ] -1
here, n = 95, p1 = 2, a1 = 5, p2 = 3, a2 = 1
multiplying compositions of a1 and a2 i get, (2^4) * (2^0) = 16
so perfect_partition(95) = 16 according to this post while answer is 112 OEIS link
If you can explain what am i missing?
2011-05-12 davidlowryduda
@Abhay in comment 1:
This is a very good point! In looking over my (original) post, I see that I misspoke. In particular, the conclusion at the end is not that the number of perfect partitions is the same as the number of compositions, but instead as the number of ordered factorizations of . I
think that I misspoke because when we only consider a prime power, there is no
difference. I have updated the post - thank you.