= Two-row Young permutation module decomposition
{title2=$M^{(n-k,k)}$}
For $0\leq k\leq n/2$, over the <complex numbers>,
$$
\boxed{M^{(n-k,k)}\cong\bigoplus_{j=0}^k S^{(n-j,j)}.}
$$
By <Young's rule>, the multiplicity is the <Kostka number> for content $(n-k,k)$. A <semistandard Young tableau> on just the entries $1,2$ has at most two rows. For shape $(n-j,j)$, every lower entry is $2$ and every entry above it is $1$; the remaining top row is uniquely fixed by the content. Such a filling exists precisely for $j\leq k$. Thus each displayed <Specht module> occurs once. A zero second part is omitted.
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