Macaulay's theorem for vector-spread algebras

Let S = K[x1,..,xn] be the standard graded polynomial ring, with K a field, and let t = (t1,..,td-1) ∈ ℤd-1 ≥, d ≥ 2, be a (d-1)-Tuple whose entries are non-negative integers. To a t-spread ideal I in S, we associate a unique ft-vector and we prove that if I is t-spread strongly stable, then there exists a unique t-spread lex ideal which shares the same ft-vector of I via the combinatorics of the t-spread shadows of special sets of monomials of S. Moreover, we characterize the possible ft-vectors of t-vector spread strongly stable ideals generalizing the well-known theorems of Macaulay and Kruskal-Katona. Finally, we prove that among all t-spread strongly stable ideals with the same ft-vector, the t-spread lex ideals have the largest Betti numbers.