= Symmetric-group characters are integer-valued
{title2=$\chi^\lambda(g)\in\mathbb Z$}
The character of a <Young permutation module> counts fixed <tabloids> and is integer-valued. <Dominance order on partitions> makes its decomposition into <Specht modules> triangular with diagonal multiplicity one. Descending induction in any order refining dominance therefore expresses each irreducible Specht character as an integer combination of permutation characters, proving integrality.
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