Find an asymptotically tight estimate for the sum $$ A_n^{k}(\lambda)= \sum_{ \substack{a_i\geq \lambda_i \\ a_1+a_2+\dots a_k=n }} \prod_{i=1}^k a_i! $$
Is the leading term going to be $$|\textrm{Number of Maximal Lambda}|(j-\lambda_1-\lambda_2-\lambda_3- \lambda_4+ \lambda_{max})!\frac{1}{\lambda_{\max}!}\prod_{i=1}^4 \lambda_i! $$
Edit: As of right now there is some discrepency as to weather this conjecture is correct or not.
This question has been asked before in the binomial situation HereHere