"Partial" factorials

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Other than double factorials and triple factorials, what is known about factorials which are missing some of their factors? For example, $1344=2 \times 3 \times 4 \times 7 \times 8$ which is $\frac{8!}{5 \times 6}$; I.e., $8!$ missing two factors - $5$ and $6$. This was encountered in studying the number of Hamiltonian circuits of a hypercube. A hypercube of dimension $4$ has $1344$ Hamiltonian circuits.