Polynomial and combinatorial analogues of Gauss congruence

The cyclic sieving phenomenon provides a link between a polynomial analogue of Gauss congruence known as $q$-Gauss congruence, and a combinatorial analogue of Gauss congruence based on sequences of cyclic group actions. We strengthen this link in two major ways: by characterising $q$-Gauss congruence via explicit formulae, and by developing a universal model for the combinatorics based on necklaces which allow beads to vary in both colour and length. This gives many novel examples of cyclic sieving involving necklaces, path walks, tubings and more. We extend the definition of Gauss congruence to sequences indexed by an arbitrary ranked semigroup, and synthesise known results into this theory.