Suppose there are \(n\) distinct numbers \(x_1,\ldots, x_n\), and \(y_1, \ldots, y_n\) is a random permutations of them. If \(\exists k\) such that \(y_k<y_i, \forall 1\le i<k\), then we say that \(y_k\) is a record (we always count \(y_1\) as a record). What is the expected number of records in \(y_1, \ldots, y_n\)?

See my neat solution here.