Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — and, true to form, there was!

$$\displaystyle \Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,\rm dt \,.$$ Complex analysis means that integrals like this one make sense over $\mathbb{C}$, so, since this function is the same as $f(n) = n!$ on $\mathbb{N}$ (shifted by one), we call it an extension of the factorial. (Is it the only complex analytic function that is an extension? No, but with a few more restrictions, we can make $\Gamma(z)$ the "only" answer to the perfect interpolation question about $n!$.)

Naïvely, I look at the first few values of the partition number $p(n)$, and want to do the same thing. But, the partitions have no simple explicit formula; that's part of their mystery!

Emory math professor Ken Ono explains the state of things in this video. (Here is the relevant paper, as well.) There are some details that I still do not understand about the paper, but in the final steps before Bruinier and Ono give their explicit formula, they introduce a method that only makes sense when applied to a natural number, so we get a very cool formula... but not something that gives us a way of thinking about $p(z)$; it's still $p(n)$.

So, my question is this: is this a fool's errand? Can we say that there will never be a meaning for something like $p(2.6)$ or $p(3-\pi i)$? I think that this could be the case, because, if such an interpretation existed, then the problem of counting partitions would be much easier than it has proven to be thus far.