Online sorting without modifications

Question

There is an array with $n$ places. There is a stream of $n$ unique numbers that arrive at a random order (permutation selected uniformly at random).

Whenever a number arrives, we must put it somewhere in the array, and we are not allowed to move it later. The goal is to have as many numbers as possible (in expectation) in their correct location in the final array. The "correct location" is defined as the location where it would appear if the numbers were sorted.

What is known about this problem? What is the best expected success rate, and what algorithm attains it?

_Notes:

• _{A related question is: What is the fastest online sorting algorithm? . It discusses a situation in which items can be moved when new items arrive, and the goal is to minimize the running time.}

• _{I am mainly interested in maximizing the expected number of the correct positions. However, it is also interesting if there is a way to improve the probability that all numbers are in the correct position, above O(1/n!)}