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An integer partition is a decomposition of a positive integer into positive integers that add up the original number. Apart from $\mathbb N_{> 0}$, does this definition apply to other numeric sets, like $\mathbb R$ or $\mathbb Q$?

nightcod3r
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I suppose you could define partitions on $\mathbb{R}$ (and $\mathbb{Q}$) similarly as you do on the positive integers. So say given a positive real number $r$, a partitioning of $r$ over $\mathbb{R}$ would be a (probably finite) collection of positive real numbers that sum to $r$. This wouldn't be as interesting, though, because given any real number, there are infinitely many ways to partition it over the reals.

What could be interesting would be to specify some weird $A \subset \mathbb{R}$ and talk about partitioning an arbitrary real number over $A$. Unfortunately, I can't think of a choice for $A$ that would yield anything nontrivial that is more interesting than partitioning over $\mathbb{R}$.

Mike Pierce
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  • Another way would be to restrict the set from where the parts are taken (e.g., reals below a given bound). – nightcod3r Mar 25 '16 at 01:03
  • @nightcod3r, yeah, but then you still get an infinite number of ways to partition anything. – Mike Pierce Mar 25 '16 at 01:06
  • What about this example: partitioning a positive rational into positive rationals $\frac{a}{b}$ where $b<k$. For $k = 20$; $\frac{3}{8}=\frac{3}{16}+\frac{3}{16}=\frac{1}{8}+\frac{1}{4}$. – nightcod3r Mar 25 '16 at 01:10
  • @nightcod3r, Suppose you want to partition $\frac{x}{y}$ over positive rational numbers with denominators less than $k$. Let $\ell$ be the least common multiple of $y$ and all positive integers less than than $k$. Then given any partitioning of $\frac{x}{y}$, we can multiply the partitioning by $\ell$ and we arrive at an integer partition of $\frac{x\ell}{y}$. I believe that this gives a one-to-one correspondence between your partitions of $\frac{x}{y}$ and integer partitions of $\frac{x\ell}{y}$, so really this isn't something new: we are just counting integer partitions again. – Mike Pierce Mar 25 '16 at 01:26
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    @night, you might be interested in Egyptian fraction decomposition. – J. M. ain't a mathematician Mar 25 '16 at 01:30
  • @MikePierce I believe you are right. – nightcod3r Mar 25 '16 at 01:34
  • @J.M., indeed, very much connected. – nightcod3r Mar 25 '16 at 01:34