Regularity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math>-invariant monomial ideals

For a polynomial ring S in n variables, we consider the natural action of symmetric group Sn on by permuting variables. an Sn-invariant monomial ideal I?S and j?0, give explicit recipe for computing modules ExtSj(S/I,S), use this to describe projective dimension regularity I. We classify ideals I that have linear free resolution, also characterize those which are Cohen–Macaulay. then two settings analyzing asymptotic behavior regularity: one where look at powers fixed I, another vary ambient examine invariant induced In first case determine generated Sn-orbit single solving integer optimization problem. second any recovering recent result Murai.