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This paper studies the numbers of minimal generators of powers of monomialideals 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 ofminimal generators of $I^2$ with $\mu(I)\geq 6$. Recently, Gasanova constructedmonomial ideals such that $\mu(I)>\mu(I^n)$ for any positive integer $n$. Inreference 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 $n$, whichprovides one of the most unexpected behaviors of the function $\mu(I^k)$. Themonomial ideals also give a peculiar example such that the Cohen-Macaulay type(or the index of irreducibility) of $R/I^n$ descends.

Certain monomial ideals whose numbers of generators of powers descend