  We investigate the cohomology rings of regular semisimple Hessenberg
varieties whose Hessenberg functions are of the form $h=(h(1),n\dots,n)$ in Lie
type $A_{n-1}$. The main result of this paper gives an explicit presentation of
the cohomology rings in terms of generators and their relations. Our
presentation naturally specializes to Borel's presentation of the cohomology
ring of the flag variety and it is compatible with the representation of the
symmetric group $\mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko.
As a corollary, we also give an explicit presentation of the
$\mathfrak{S}_n$-invariant subring of the cohomology ring.
