The $e$-expansion of integral form Macdonald polynomials

In this paper, we investigate the expansion of the integral form Macdonald polynomials $J_μ(x;q,t)$ in the elementary symmetric function basis. Motivated by the connection between chromatic symmetric functions and LLT polynomials, we conjecture the $e$-positivity of a normalization of $J_μ(x;q,t)$. We verify the conjecture when $μ$ is either a hook or a two-column partition. In the hook case, we express the $e$-coefficients as weighted sums over compositions. In the two-column case, we derive explicit formulas for the $e$-coefficients.