TITLE: Why does McMahon formula look like the inclusion-exclusion principle?
QUESTION [31 upvotes]: The McMahon formula 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.  
Perhaps there is a more general explanation via Gelfand-Tsetlins or something?

(source: microsoft.com)

REPLY [2 votes]: This is not an answer, but seeing as this is a fresh post, I would like to add:

These are the $20\; 2\times2\times2$ plane partitions. Note there are $4$ of each color when viewed as hexagons.
If we let $a=1$, we get:
$$\dbinom{b+c}{c}=\prod_{i=1}^b\prod_{j=1}^c \dfrac{i+j}{i+j-1}=\prod_{i=1}^c \dfrac{i+b}{i}$$
A few more links:

Mathworld
Wikipedia
DLMF

Note the I-E principle is divisive and not subtractive.