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We establish a relation between the Sprague-Grundy function $\text{sg}$ of a$p$-saturation of Welter's game and the degrees of the ordinary irreduciblerepresentations of symmetric groups. In this game, a position can be viewed asa partition $\lambda$. Let $\rho^\lambda$ be the irreducible representation of$\text{Sym}(|\lambda|)$ indexed by $\lambda$. For every prime $p$, we show thefollowing results: (1) the degree of $\rho^\lambda$ is prime to $p$ if and onlyif $\text{sg}(\lambda) = |\lambda|$; (2) the restriction of $\rho^\lambda$ to$\text{Sym}(\text{sg}(\lambda))$ has an irreducible component with degree primeto $p$. Further, for every integer $p$ greater than 1, we obtain an explicitformula for $\text{sg}(\lambda)$.

p-Saturations of Welter's Game and the Irreducible Representations of Symmetric Groups