  Strongly stable monomial ideals are important in algebraic geometry,
commutative algebra, and combinatorics. Prompted, for example, by combinatorial
approaches for studying Hilbert schemes and the existence of maximal total
Betti numbers among saturated ideals with a given Hilbert polynomial, in this
note we present three algorithms to produce all strongly stable ideals with
certain prescribed properties: the saturated strongly stable ideals with a
given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert
polynomial, and the saturated strongly stable ideals with a given Hilbert
function. We also establish results for estimating the complexity of our
algorithms.
