Abstract
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 \(\mu (I)\ge 6\). 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.
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