The [McMahon formula](http://arxiv.org/abs/math/9808017) for the number of tilings of an $a \times b \times c$ hexagon by lozenges:

$$ \Big[H(a)H(b)H(c)\Big] \Big[H(a+b)H(b+c)H(c+a)\Big]^{-1} \Big[H(a+b+c)\Big]$$

looks oddly like the inclusion-exclusion formula:

$$ |A \cup B \cup C| = |A|+|B|+|C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|$$

Here $H(a) = 1! 2! \dots a!$ is the [hyperfactorial](https://en.wikipedia.org/wiki/Factorial#Hyperfactorial).  

Perhaps there is a more general explanation via [Gelfand-Tsetlins](http://ncatlab.org/nlab/show/Gelfand-Tsetlin+basis) or something?

<img src="http://research.microsoft.com/en-us/um/people/cohn/Graphics/hexagon.gif" width="200">