I would like to know if anyone knows where I can find a proof for the equivalent hook-length formula
$$f^{\lambda}=\frac{n! \cdot \prod_{i<j}(l_i-l_j)}{l_1! \cdot l_2! \cdot ... \cdot l_k!}$$
where if $\lambda=(\lambda_1 \geq \lambda_2\geq ...\geq \lambda_k \geq 0)$ is a partition, then $l_i=\lambda_i-i + k$ for all $i$.
I have seen the formula in the book Young Tableaux of William Fulton.
A Course in Combinatorics by Van Lint and Wilson has a proof (Theorem 15.10).
Enumerative Combinatorics Vol. 2 by Richard Stanley has a proof (see Lemma 7.21.1, equation 7.101)
A Course in Enumeration by Martin Aigner has a proof (around page 388)
Symmetric Functions and Hall Polynomials by Ian Macdonald I believe has a proof, but I can't check right now.
(5.17.1) in Introduction to Representation Theory by Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina with historical interludes by Slava Gerovitch