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We study type $A$ nilpotent Hessenberg varieties equipped with a natural$S^1$-action using techniques introduced by Tymoczko, Harada-Tymoczko, andBayegan-Harada, with a particular emphasis on a special class of nilpotentSpringer varieties corresponding to the partition $\lambda= (n-2,2)$ for $n\geq 4$. First we define the adjacent-pair matrix corresponding to any fillingof a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. Usingthe adjacent-pair matrix we make more explicit and also extend some statementsconcerning highest forms of linear operators in previous work of Tymoczko.Second, for a nilpotent operator $N$ and Hessenberg function $h$, we constructan explicit bijection between the $S^1$-fixed points of the nilpotentHessenberg variety $\Hess(N,h)$ and the set of $(h,\lambda_N)$-permissiblefillings of the Young diagram $\lambda_N$. Third, we use poset pinball, thecombinatorial game introduced by Harada and Tymoczko, to study the$S^1$-equivariant cohomology of type $A$ Springer varieties$\mathcal{S}_{(n-2,2)}$ associated to Young diagrams of shape $(n-2,2)$ for$n\geq 4$. Specifically, we use the dimension pair algorithm forBetti-acceptable pinball described by Bayegan and Harada to specify a subset ofthe equivariant Schubert classes in the $T$-equivariant cohomology of the flagvariety $\mathcal{F}\ell ags(\C^n)$ which maps to a module basis of$H^*_{S^1}(\mathcal{S}_{(n-2,2)})$ under the projection $H^*_T(\mathcal{F}\ellags(\C^n)) \to H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. Our pinball module basis isnot poset-upper-triangular; this is the first concrete such example in theliterature. A consequence of our proof is that there exists a simple andexplicit change of basis which transforms our basis to a poset-upper-triangularmodule basis for $H^*_{S^1}(\mathcal{S}_{(n-2,2)})$. We close with openquestions for future work.

Poset pinball, highest forms, and (n-2,2) Springer varieties