I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural numbers (can be assumed all different) and $\sigma$ runs over the full permutation group on $\{1,...,n\}$.
Does this resemble anything simpler? I want any kind of additional information. To ask something slightly more concrete: expanding this into powers of $t$, coefficients are obviously symmetric functions of $k_1,\ldots,k_n$; they look quite impenetrable. For example, $$ F_{k_1,k_2,k_3,k_4}(t) = 24-90s_1t^2 + \left(\frac{177}2s_1^2+96s_2\right)t^4 - \left(\frac{163}4s_1^3 + \frac{233}2s_1s_2 + 21s_3 \right)t^6 + \text{(more and more horrible fractions)}, $$ where $s_1, s_2, \ldots$ are the elementary symmetric functions of the $k_i$'s, $s_1=k_1+k_2+k_3+k_4$, $s_2=k_1k_2+k_1k_3+...+k_3k_4$, etc. How to treat this?
What would be ideal for me is a decomposition into a product of some nice terms (preferably into sums of sines or cosines, something like that).