Nonincreasing depth functions of monomial ideals

Given a nonincreasing function $f : \mathbb{Z}_{\geq 0} \setminus \{ 0 \} \to \mathbb{Z}_{\geq 0}$ such that (i) $f(k) - f(k+1) \leq 1$ for all $k \geq 1$ and (ii) if $a = f(1)$ and $b = \lim_{k \to \infty} f(k)$, then $|f^{-1}(a)| \leq |f^{-1}(a-1)| \leq \cdots \leq |f^{-1}(b+1)|$, a system of generators of a monomial ideal $I \subset K[x_1, \ldots, x_n]$ for which ${\rm depth} S/I^k = f(k)$ for all $k \geq 1$ is explicitly described. Furthermore, we give a characterization of triplets of integers $(n,d,r)$ with $n > 0$, $d \geq 0$ and $r > 0$ with the properties that there exists a monomial ideal $I \subset S = K[x_1, \ldots, x_n]$ for which $\lim_{k \to \infty} {\rm depth} S/I^k = d$ and ${\rm dstab}(I) = r$, where ${\rm dstab}(I)$ is the smallest integer $k_0 \geq 1$ with ${\rm depth} S/I^{k_0} = {\rm depth} S/I^{k_0+1} = {\rm depth} S/I^{k_0+2} = \cdots$.