Betti tables of monomial ideals fixed by permutations of the variables

Let S_n be a polynomial ring with n variables over a field and {I_n}_n≥1 a chain of ideals such that each I_n is a monomial ideal of S_n fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of I_n for all large intergers n from the Z^mgraded Betti tables of I_m for some small integers m. Our main result shows that the projective dimension and the regularity of I_n eventually become linear functions on n, confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.