  Regular nilpotent Hessenberg varieties are subvarieties of full flag
varieties, while regular nilpotent partial Hessenberg varieties are
subvarieties of partial flag varieties. In this manuscript we first provide a
summand formula and a product formula for the Poincar\'e polynomial of regular
nilpotent partial Hessenberg varieties. It is well-known that there is an
isomorphism between the cohomology rings of partial flag varieties and the
invariants in the cohomology rings of full flag varieties under an action of a
certain Weyl group by Bernstein-Gelfand-Gelfand. We generalize this result to
regular nilpotent partial Hessenberg varieties. More concretely, we give an
isomorphism between the cohomology rings of regular nilpotent partial
Hessenberg varieties and the invariant subrings of the cohomology rings of
regular nilpotent Hessenberg varieties under the certain Weyl group action.
Furthermore, we provide a description of the cohomology rings for regular
nilpotent partial Hessenberg varieties in terms of the invariants in the
logarithmic derivation modules of ideal arrangements, which is a genelarization
of the result by Abe-Masuda-Murai-Sato with the author.
