Symmetric iterated Betti numbers

We define a set of invariants of a homogeneous ideal I in a polynomial ring called the symmetric iterated Betti numbers of I. We prove that for IΓ, the Stanley–Reisner ideal of a simplicial complex Γ, these numbers are the symmetric counterparts of the exterior iterated Betti numbers of Γ introduced by Duval and Rose, and that the extremal Betti numbers of IΓ are precisely the extremal (symmetric or exterior) iterated Betti numbers of Γ. We show that the symmetric iterated Betti numbers of an ideal I coincide with those of a particular reverse lexicographic generic initial ideal Gin(I) of I, and interpret these invariants in terms of the associated primes and standard pairs of Gin(I). We close with results and conjectures about the relationship between symmetric and exterior iterated Betti numbers of a simplicial complex.