The Pieri formula gives a decomposition
$\textrm{Sym}^d V \otimes \textrm{Sym}^d V = \bigoplus \mathbb{S}_{(d+a, d-a)}V,$
the sum over $0\leq a \leq d$. The left-hand side decomposes into a direct sum of $\textrm{Sym}^2(\textrm{Sym}^d V)$ and $\bigwedge^2(\textrm{Sym}^d V)$. Show that, in fact,
$\textrm{Sym}^2(\textrm{Sym}^d V)=\mathbb{S}_{(2d,0)}V\bigoplus \mathbb{S}_{(2d-2,2)}V\bigoplus \mathbb{S}_{(2d-4,4)}V \bigoplus...$
and
$\bigwedge^2(Sym^d V)=\mathbb{S}_{(2d-1,1)}V\bigoplus \mathbb{S}_{(2d-3,3)}V\bigoplus \mathbb{S}_{(2d-5,5)}V \bigoplus...$