Functions with optional argument named FirstBetti :
bettiTablesWithHilbertFunction(...,FirstBetti=>...) -- see bettiTablesWithHilbertFunction -- list or tally all Betti tables that can be obtained from monomial ideals with a particular (partial or complete) Hilbert function
monomialIdealsWithHilbertFunction(...,FirstBetti=>...) -- see monomialIdealsWithHilbertFunction -- find all monomial ideals in a polynomial ring with a particular (partial or complete) Hilbert function