  In this manuscript we develop the theory of poset pinball, a combinatorial
game recently introduced by Harada and Tymoczko for the study of the
equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada
and Tymoczko also prove that in certain circumstances, a successful outcome of
Betti poset pinball yields a module basis for the equivariant cohomology ring
of the GKM-compatible subspace. Our main contributions are twofold. First we
construct an algorithm (which we call the dimension pair algorithm) which
yields the result of a successful outcome of Betti poset pinball for any type
$A$ regular nilpotent Hessenberg and any type $A$ nilpotent Springer variety,
considered as GKM-compatible subspaces of the flag variety $\Flags(\C^n)$. The
definition of the algorithm is motivated by a correspondence between Hessenberg
affine cells and certain Schubert polynomials which we learned from Erik Insko.
Second, in the special case of the type $A$ regular nilpotent Hessenberg
varieties specified by the Hessenberg function $h(1)=h(2)=3$ and $h(i) = i+1$
for $3 \leq i \leq n-1$ and $h(n)=n$, we prove that the pinball result coming
from the dimension pair algorithm is poset-upper-triangular; by results of
Harada and Tymoczko this implies the corresponding equivariant cohomology
classes form a $H^*_{S^1}(\pt)$-module basis for the $S^1$-equivariant
cohomology ring of the Hessenberg variety.
