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Given a finite discrete set $X \subset \mathbb{R}$ with cardinality $k$.

The are $2^k$ subsets. For any given subset, there is a median associated with it (Define the median of $\emptyset$ as 0).

Wondering if there are any smart ways to get the $2^k$ medians.

peng yu
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  • I don't know a smart way to get the medians, but they are not $2^k$. I generated several sets of $10$ "real" numbers, with $2^{10}$ subsets and their distinct median are always $56$. It looks like an interesting phenomenon – Raffaele Feb 02 '21 at 18:14
  • I tried other cardinalities and got an amazing result. The number of distinct medians of the subsets of $n$ numbers is $$\frac{1}{2} \left(n^2+n+2\right)$$ – Raffaele Feb 02 '21 at 18:16
  • interesting, how did you get that number? would love to hear! – peng yu Feb 02 '21 at 18:24
  • https://math.stackexchange.com/questions/4009926/median-of-the-subsets-of-a-finite-set-of-real-numbers – Raffaele Feb 02 '21 at 18:27
  • ah right, so you guessed the formula from observation – peng yu Feb 02 '21 at 18:29
  • I am not clever enough to prove it :) – Raffaele Feb 02 '21 at 18:30
  • i mean, it would be nicer, if you mention this part explicitly. it could be some other function that has a similar form? it could mislead folks trying to prove with the specific function. – peng yu Feb 02 '21 at 18:32
  • It is a conjecture. If it is false, contributor will soon find out. People are very very clever here – Raffaele Feb 02 '21 at 18:34
  • lol, ok, really interesting conjecture! – peng yu Feb 02 '21 at 18:35
  • lol, the smart proof is a bit "disappointing" – peng yu Feb 02 '21 at 18:46

1 Answers1

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The possible medians are:

  • any of the $k$ values
  • (if you define the median of an even-sized set as halfway between the two most central values) any mean of two of the $k$ values
  • $0$ for the empty set

That makes $k+{k \choose 2}+1 = \frac12k^2 +\frac12k+1$ potentially distinct values for subset medians and explains how to find them; it may turn out that there are some duplicates.

Henry
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