Lexsegment ideals and their \(h\)-polynomials (Q2000794)

Let \(S=K[x_1, \dots, x_n]\) be the polynomial ring in \(n\) variables over a field \(K\) with its standard grading. For any homogeneous ideal \(I\subset S\) with \(\dim S/I = d\), the Hilbert series of \(S/I\) is of the form \(h_{S/I}(\lambda)/(1-\lambda)^d\), where \(h_{S/I}(\lambda)=h_0+h_1\lambda+ h_2\lambda^2+ \cdots +h_s\lambda^s\) with \(h_s\neq 0\) is the \(h\)-polynomial of \(S/I\). A monomial ideal \(I\subset S\) is called lexsegment if for any monomial \(u\in I\) and any monomial \(v\in S\) with \(\mathrm{deg } u=\mathrm{deg } v\) and \(v >_{\mathrm{lex}} u\) one has \(v\in I\). In the paper under review, for any pair of integers \(r, s\geq 1\), the authors construct lexsegment ideal \(I\) over at most \(\max\{r, s\} + 2\) variables, satisfying \(\mathrm{reg}(S/I)=r\) and \(\mathrm{deg }h_{S/I}(\lambda)=s\). The authors close their article with the following interesting question: Question. Find all possible sequences \((d, e, r, s)\in \mathbb{Z}_{\geq 0}\) with \(d\geq e\geq 0, r\geq 1, s\geq 1\) and \(s-r\leq d-e\) for which there exists a homogeneous ideal \(I\subset S=K[x_1, \dots, x_n]\) with \(n\gg 0\) satisfying \[ \dim(S/I)=d, \ \mathrm{depth}(S/I)=e, \ \mathrm{reg}(S/I)=r, \ \mathrm{deg } h_{S/I}(\lambda)=s. \]