Nonincreasing depth functions of monomial ideals

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