# What is the probability pairs of sums are distinct?

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:

$$\prod_{i=1}^{n-1}\left(1-\frac{i}{s}\right)$$

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.