Exterior-algebra Hilbert-series conjecture

Prove or disprove that, for every n, the Hilbert series of the exterior algebra E_n on n generators modulo two generic quadratic forms equals 1+c(n,1)t+c(n,2)t^2+⋯+c(n,⌊n/2⌋)t^{⌊n/2⌋}, where c(n,s) counts the specified lattice paths.

Background

The paper introduces two conjectures from Cruz and Iarrobino relating lattice-path counts to Hilbert series. It proves the analogous conjecture for powers of general linear forms, and the cited relationship implies the exterior-algebra conjecture in even numbers of generators if the powers-of-linear-forms conjecture holds universally. However, the paper does not establish the exterior-algebra formula for all n; its unresolved status is therefore retained in the stated conjecture.

References

The Hilbert series of $E_n/(f,g)$ is equal to $1+c(n,1)t+c(n,2)t2 + \cdots + c(n, \lfloor \frac{n}{2} \rfloor) t{\lfloor \frac{n}{2} \rfloor}$.

— On the Hilbert series of ideals generated by general linear forms  (2608.22823 - Boij et al., 24 Aug 2026) in Conjecture 5.1, Section 5, “Two generic quadratic forms in the exterior algebra”