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 equivalent to the Alon-Tarsi conjecture. For an attempt at explaining why (I think) this conjecture is important, see my answer at this MO post:
What are the current breakthroughs of Geometric Complexity Theory?