I became irritated when quick sort was claimed be O(n^2). Not only is that not true in practice, it's just wrong for all but the simplest implementations of the algorithm.

With minor tweaks to the selection of partitions, quick sort can be made O(n lg n). Specifically, if combined with linear time median finding, we can pick perfect partitions every time. (Of course, this algorithm is painfully slow in practice).

All this said, the idea of analytic combinatorics is interesting for a deeper understanding of algorithm performance. It quantifies behavior that good engineers understand is swept under the rug by Big-O notation.

Of course, if you really want good performance, you'll have to understand the algorithm beyond just it's combinatorial properties -- caching locality and coherence, branch mispredictions, function call-overhead etc. play an enormous role when it comes time to really make code fast.

To write fast code, you need the whole picture.