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 Jul 2 awarded Curious Feb 5 comment Where is the mistake in this argument about number of different balls with replacement Thanks for the good write-up! I've got a better intuition of the situation now. Feb 5 accepted Where is the mistake in this argument about number of different balls with replacement Feb 5 comment Where is the mistake in this argument about number of different balls with replacement Thanks for the answer. Yes that is my intuition too. However I fail to see how the events $D>0$ and $E_k$ are different. If $D>0$, if and only if there exists a $k$ such that $E_k$ is true. Feb 5 asked Where is the mistake in this argument about number of different balls with replacement Jan 31 revised Choosing the vector that minimizes this sum related to the rearrangement inequality deleted 91 characters in body Jan 30 comment Choosing the vector that minimizes this sum related to the rearrangement inequality @PeterKošinár: I modified the statement slightly following our discussion and my experiments, and I think it is more correct now. Jan 30 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 153 characters in body Jan 28 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 23 characters in body Jan 28 comment Choosing the vector that minimizes this sum related to the rearrangement inequality @PeterKošinár: Yes it seems there are problems with small $n$. My intention is to work with big values of $n$, but this example is degenerated (but correct and helpful). The intuition here is that it is preferable, in order to minimize the sum, to pair big $x$'s with small $y$'s, in the same line of thought as the rearrangement inequality. In this case the sum cuts short too early for this strategy to prove useful. Perhaps the conjecture is only true for $n\to\infty$? (or perhaps is it not true at all) Jan 27 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 2 characters in body Jan 26 comment Choosing the vector that minimizes this sum related to the rearrangement inequality Thank you, I corrected the example. Regarding your example, we have $S([0]) = 1 < S([1]) = 2$, and the conjectured condition proposes indeed $b^* = [0]$. Jan 26 revised Choosing the vector that minimizes this sum related to the rearrangement inequality edited body Jan 25 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 12 characters in body Jan 25 revised Choosing the vector that minimizes this sum related to the rearrangement inequality edited body Jan 25 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 258 characters in body Jan 24 revised Recurrence equation similar to a geometric progression added 2 characters in body Jan 24 revised Choosing the vector that minimizes this sum related to the rearrangement inequality deleted 2 characters in body Jan 24 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 19 characters in body Jan 23 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 6 characters in body