Gin and lex of certain monomial ideals

Let A = K[x₁, . . ., x_n] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg x_i = 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) β_k(I) < β_k(Gin(I)) for all k < i, (ii) β_i(I) = β_i(Gin(I)), (iii) β_ℓ(Gin(I)) < β_ℓ(Lex(I)) for all ℓ < j and (iv) β_j(Gin(I))=β_j(Lex(I)), where Gin(I) is the generic initial ideal of I with respect to the reverse lexicographic order induced by x₁ > . . . > x_n and where Lex(I) is the lexsegment ideal with the same Hilbert function as I.