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In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem byusing a Sylvester-style bijection. In this paper, we introduce two families ofcolored partitions, flat and regular partitions, and generalize the bijectionof Keith and Xiong to these partitions. We then state two results, the first atdegree one, where partitions have parts with primary colors, and the secondresult at degree two for secondary-colored partitions, using the result of thefirst paper of this series. These results allow us to easily retrieve theFrenkel-Kac character formulas of level one standard modules for the type$A_{2n}^{(2)}, D_{n+1}^{(2)}$ and $B_n^{(1)}$, and also to make the connectionbetween the result stated in paper one and the representation theory.

Weighted words at degree two, II: flat partitions, regular partitions, and application to level one perfect crystals