For $s \geq n$, sample with replacement $n$ integers $A_i$ from the range $\{1,\dots,s\}$ uniformly and independently. The probability that all the $A_i$ are distinct is:


Now instead consider the sums of pairs of consecutive values. That is $$Z_1 = A_1+A_2, Z_2 = A_2+A+3, Z_3 = A_3+A_4, \dots$$

What is the probability that all of the $Z_i$ are distinct?

Numerically it seems that if $s = n^2$ the probability is approximately $0.75$. I don't know how true this mathematically is however.


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