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The permutation character ${\xi }^{\mu }(g)$ is the number of $\mu$-tabloids fixed by $g$, since the trace of a permutation matrix counts fixed basis elements. It is therefore an integer. Arrange the partitions in decreasing lexicographic order. If $\lambda$ strictly dominates $\mu$, the first part at which they differ is larger for $\lambda$, so this ordering places $\lambda$ before $\mu$.

For the first partition $(n)$, there is just one tabloid, and ${\chi }^{(n)}={\xi }^{(n)}=1_{S_n}$ is integer-valued. At a general partition $\mu$, the decomposition in (iii) can be rearranged asBy induction, all characters in the sum are already integer-valued; the permutation character and all coefficients are also integral. Thus ${\chi }^{\mu }$ is integer-valued. Finite induction proves that symmetric-group characters are integer-valued:The diagonal coefficient one is essential: it lets us solve for ${\chi }^{\mu }$ without division, preserving integrality.