  Let $S_n$ be a polynomial ring with $n$ variables over a field and
$\{I_n\}_{n \geq 1}$ 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 $\mathbb Z^m$-graded Betti table of $I_m$ for some
integer $m$. 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 R\"omer.
