Let A = K[x1, . . ., xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βℓ(Gin(I)) < βℓ(Lex(I)) for all ℓ < j and (iv) βj(Gin(I))=βj(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x1 > . . . > xn and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.
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