Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet connects the well-known Stanley-Stembridge conjecture in combinatorics to the dot action of the symmetric group $S_n$ on the cohomology rings $H^*(Hess(S,h))$ of regular semisimple Hessenberg varieties. In particular, in order to prove the Stanley-Stembridge conjecture, it suffices to construct (for any Hessenberg function $h$) a permutation basis of $H^*(Hess(S,h))$ whose elements have stabilizers isomorphic to Young subgroups. In this manuscript we give several results which contribute toward this goal. Specifically, in some special cases, we give a new, purely combinatorial construction of classes in the $T$-equivariant cohomology ring $H^*_T(Hess(S,h))$ which form permutation bases for subrepresentations in $H^*_T(Hess(S,h))$. Moreover, from the definition of our classes it follows that the stabilizers are isomorphic to Young subgroups. Our constructions use a presentation of the $T$-equivariant cohomology rings $H^*_T(Hess(S,h))$ due to Goresky, Kottwitz, and MacPherson. The constructions presented in this manuscript generalize past work of Abe-Horiguchi-Masuda, Chow, and Cho-Hong-Lee.