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 Jul2 awarded Curious Feb5 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. Feb5 accepted Where is the mistake in this argument about number of different balls with replacement Feb5 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. Feb5 asked Where is the mistake in this argument about number of different balls with replacement Jan31 revised Choosing the vector that minimizes this sum related to the rearrangement inequality deleted 91 characters in body Jan30 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. Jan30 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 153 characters in body Jan28 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 23 characters in body Jan28 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) Jan27 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 2 characters in body Jan26 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 $b^* = [0]$. Jan26 revised Choosing the vector that minimizes this sum related to the rearrangement inequality edited body Jan25 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 12 characters in body Jan25 revised Choosing the vector that minimizes this sum related to the rearrangement inequality edited body Jan25 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 258 characters in body Jan24 revised Recurrence equation similar to a geometric progression added 2 characters in body Jan24 revised Choosing the vector that minimizes this sum related to the rearrangement inequality deleted 2 characters in body Jan24 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 19 characters in body Jan23 revised Choosing the vector that minimizes this sum related to the rearrangement inequality added 6 characters in body