For every n and d there is a monomial ideal in n variables with |G(I)| > |G(I^i)| for all 2 ≤ i ≤ d.
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Monomial ideals with arbitrarily high tiny powers in any number of variables
For every n and d there is a monomial ideal in n variables with |G(I)| > |G(I^i)| for all 2 ≤ i ≤ d.