The saturation number of $cb$-bounded stable monomial ideals and their powers

Let $S=K[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. In this paper, we compute the socle of $cb$-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated $cb$-bounded strongly stable ideals. We also provide explicit formulas for the saturation number $sat(I)$ of Veronese type ideals $I$. Using this formula, we show that $sat(I^k)$ is quasi-linear from the beginning and we determine the quasi-linear function explicitly.