From Dummit Foote Chapter 11.5: pages 452 and 453. I believe I was able to reprove these with $Alt$ instead of $\frac{1}{k!} Alt$. Maybe I made a mistake.
What's the difference between $Alt$ and $\frac{1}{k!} Alt$? Is it perhaps merely a conventional thing like the $a_0$ vs $\frac{a_0}{2}$ in Fourier series?
We can think of Alt as a projection map that maps any given covariant k-tensor to an alternating covariant k-tensor.
If $\alpha$ is already an alternating covariant k-tensor, then it looks elegant if Alt $\alpha = \alpha$ ( That is Alt should map alternating covariant k-tensors to themselves)
However, according to the definition of Alternation in Dumme and Foote, Alt $\alpha = (k!)\alpha $.
So, we introduce a new projection map, $\frac{1}{k!} Alt $, that naturally maps alternating covariant k-tensors to alternating covariant k-tensors (themselves).
That is $ \frac{1}{k!}$ Alt $\alpha=\alpha $.
So $ \frac{1}{k!}$ can be thought of as a normalization for elegance. It is elegant to define an map (operator) that takes alternating covariant k-tensors to themselves, and non-alternating covariant k-tensors to alternating covariant k-tensors.