  We propose a variation of the classical Hilbert scheme of points - the double
nested Hilbert scheme of points - which parametrizes flags of zero-dimensional
subschemes whose nesting is dictated by a Young diagram. Over a smooth
quasi-projective curve, we compute the generating series of topological Euler
characteristic of these spaces, by exploiting the combinatorics of reversed
plane partitions. Moreover, we realize this moduli space as the zero locus of a
section of a vector bundle over a smooth ambient space, which therefore admits
a virtual fundamental class.
  We apply this construction to the stable pair theory of a local curve, that
is the total space of the direct sum of two line bundles over a curve. We show
that the invariants localize to virtual intersection numbers on double nested
Hilbert scheme of points on the curve, and that the localized contributions to
the invariants are controlled by three universal series for every Young
diagram, which can be explicitly determined after the anti-diagonal restriction
of the equivariant parameters. Under the anti-diagonal restriction, the
invariants are matched with the Gromov-Witten invariants of local curves of
Bryan-Pandharipande, as predicted by the MNOP correspondence.
  Finally, we discuss $K$-theoretic refinements \`a la Nekrasov-Okounkov.
