We examine the ideal I= (x₁², , xₙ², (x₁++xₙ) ²) in the polynomial ring Q=kx₁, , xₙ, where k is a field of characteristic zero or greater than n. We also study the Gorenstein ideal G linked to I via the complete intersection ideal (x₁², , xₙ²). We compute the Betti numbers of I and G over Q when n is odd and extend known computations when n is even. A consequence is that the socle of Q/I is generated in a single degree (thus Q/I is level) and its dimension is a Catalan number. We also describe the generators and the initial ideal with respect to reverse lexicographic order for the Gorenstein ideal G.
Diethorn et al. (Fri,) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: