Addendum — Vector invariants of permutation groups in characteristic zero

This note proves that rings of vector invariants of permutation groups acting on an [Formula: see text]-dimensional vector space in characteristic zero are generated by its invariants of degree less than or equal to [Formula: see text]. This improves our previous result [F. Reimers and M. Sezer, Vector invariants of permutation groups in characteristic zero, Internat. J. Math. 35(3) (2024) 2350111] and fully extends Göbel’s theorem [M. Göbel, Computing bases for rings of permutation-invariant polynomials, J. Symbolic Comput. 19(4) (1995) 285–291] to the case of vector invariants.