Gin and Lex of certain monomial ideals

Let $A=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ denote the polynomial ring in $n$ variables over a field $K$ of characteristic 0 with each $\mathrm{deg}{x}_i=1$. Given arbitrary integers $i$ and $j$ with $2\le i\le n$ and $3\le j\le n$, we will construct a monomial ideal $I\subset A$ such that (i) $\beta_k(I) < \beta_k(\Gin(I))$forall$k < i$,(ii)$\beta_i(I) = \beta_i(\Gin(I))$,(iii)$\beta_\ell(\Gin(I)) < \beta_\ell(\Lex(I))$forall$\ell < j$and(iv)$\beta_j(\Gin(I)) = \beta_j(\Lex(I))$,where$\Gin(I)$ is the generic initial ideal of $I$ with respect to the reverse lexicographic order induced by $x_1 >

… > x_n$andwhere$\Lex(I)$ is the lexsegment ideal with the same Hilbert function as $I$.