Fluctuations in depth and associated primes of powers of ideals

We count the numbers of associated primes of powers of ideals as defined in $\href{https://dx.doi.org/10.1007/s11512-013-0184-1}{[2]}$. We generalize those ideals to monomial ideals $\operatorname{BHH}(m, r, s)$ for $r \geq 2, m, s \geq1$; we establish partially the associated primes of powers of these ideals, and we establish completely the depth function of quotients by powers of these ideals: the depth function is periodic of period $r$ repeated $m$ times on the initial interval before settling to a constant value. The number of needed variables for these depth functions are lower than those from general constructions in $\href{https://doi.org/10.1090/proc/15083}{[6]}$.