Certain monomial ideals whose numbers of generators of powers descend

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $mu (I^2)ge 9$ for the number of minimal generators of $I^2$ with $mu(I)geq 6$. Recently, Gasanova constructed monomial ideals such that $mu(I)>mu(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $mu(I)>mu(I^2)>cdots >mu(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $mu(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends.