Combinatorial? Seriously? The difference between exponential and cominatorial is essentially purely theoretical when it comes to computational complexity. In practice, even the difference between n and n log n is barely relevant, and the exponential and combinatorial are the exponentiation thereof. Nobody ever, ever deals with processes large enough where the difference both matters, and the process completes. Even a tiny scaling factor in the exponent and a moderate base means the point at which a unscaled combinatorial passes it will take more steps that you might as well be waiting for the heat death of the universe.
But even without any scaling factors, don't underestimate the size of the numbers involved. If you had two threads, one simply counting up to some n! upper bound, and the other up to 10^n - how long do you think the minimum problem size would take for the "slow" combinatorial to actually be slower than the exponentiation? Hint: at 4GHz and 1 op/cycle... Let's just say I'm not holding my breath that our species will still be around then. And even a tiny scaling factor to the mix...
I mean, if you're trying to estimate computational complexity at least, this nuance seems pointless.
But quadratic vs. exponential really matters, with plausible parameters.
With n = 26, the combinatorial process takes 4 times as long. 4 x 10^26 "counts". Googles SHA1 collision attack two years ago used roughly 10^19 computations. So we're off by a factor of 10^7.
I'm curious if the rise of quantum computers will make the differences between exponential and combinatorial meaningful in a practical sense.
"Exponential" is indeed misused.
Half the time I hear it, it is lower, like quadratic or cubic. The other half the time is higher, like combinatorial.