Not exactly the most frequent mathematical object in the literature, however, here is an interesting instance where this quantity occurs.

Take the the statistical average of $\prod_{i,j} a_{ij}$ over a special unitary $n\times n$ matrix $A$ chosen uniformly at random (i.e., Haar-distributed). Showing this expectation is nonzero for $n$ even is <a href="https://www.sciencedirect.com/science/article/pii/S0012365X15000473">equivalent to the Alon-Tarsi conjecture</a>.
For an attempt at explaining why (I think) this conjecture is important, see my answer at this MO post:

https://mathoverflow.net/questions/277408/what-are-the-current-breakthroughs-of-geometric-complexity-theory