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