= Vershik linear relations
{c}
Let $M(\mu,\lambda)$ be the multiplicity of $V^\mu$ in the complex <Young permutation module> $M^\lambda$. For $\lambda\vdash n$ and $\rho\vdash n-1$,
$$
\sum_{\mu:\rho\nearrow\mu}M(\mu,\lambda)
=\sum_{i:\lambda_i>0}M(\rho,\operatorname{sort}(\lambda-e_i)).
$$
Here $\nearrow$ means adding one cell, and zero parts are omitted. Equal row lengths contribute separately on the right. Restrict $M^\lambda$ to $S_{n-1}$ and separate the <tabloid> orbits by the row containing $n$ to obtain the right side. Decompose into irreducibles and use the <restriction branching rule for a symmetric group> to obtain the left side. Equating multiplicities proves the formula.
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