We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{μ\subseteq [K\times N]}(-1)^{|μ|}s_μ\{x\}s_{μ'}\{y\}. \end{align*} As an application of the interpretation, we obtain an expansion of $\prod^\infty_{i=1}(1-q^i)^{i-1}$ in terms of half plane partitions.