Prescribed Initial Behavior of $\mu (I^k)$

It is well known that for every graded ideal $I$, the numbers of minimal generators $\mu (I^k)$ of its powers of $I$ are eventually increasing. However, its initial behavior can be surprisingly flexible. We prove that any prescribed finite pattern of increases, decreases, and equalities can occur among the first differences $\mu (I^{k+1})-\mu (I^k)$ of a suitable monomial ideal $I$ in $K[x,y]$. This provides a broad positive answer to a previously posed sign-realization problem and unifies several known constructions exhibiting unusual behavior of $\mu (I^k)$. Moreover, as a consequence, analogous sign-realization results are obtained for the index of reducibility of powers of $\mathfrak{m}$-primary ideals in two-dimensional regular local rings.