  Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a
field $K$ of characteristic 0 with each $\deg x_i = 1$. Given arbitrary
integers $i$ and $j$ with $2 \leq i \leq n$ and $3 \leq j \leq n$, we will
construct a monomial ideal $I \subset A$ such that (i) $\beta_k(I) <
\beta_k(\Gin(I))$ for all $k < i$, (ii) $\beta_i(I) = \beta_i(\Gin(I))$, (iii)
$\beta_\ell(\Gin(I)) < \beta_\ell(\Lex(I))$ for all $\ell < j$ 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 $x_1 >
>... > x_n$ and where $\Lex(I)$ is the lexsegment ideal with the same Hilbert
function as $I$.
