In 2024, Hikita showed that the chromatic symmetric functions of incomparability graphs of (3+1)-free posets expand with positive coefficients in the basis of elementary symmetric functions. This result resolved the long-standing Stanley--Stembridge conjecture. Finding a combinatorial interpretation of the e-coefficients remains a major open problem. In this talk I will define powerful and strong P-tableaux, and use them to describe combinatorial interpretations of various cases of the e-coefficients. We conjecture that strong P-tableaux give lower bounds for the e-coefficients of chromatic symmetric functions. Additionally, we show how Hikita's theorem relates to strong P-tableaux and the Shareshian--Wachs inversion statistic.