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The generating functions for the Wilson loops in the symmetric andantisymmetric representations of the gauge group $U(N)$ are expressed in termsof the connected correlators of multiply-wound Wilson loops, using ingredientsfrom the representation theory of the symmetric group. This provides a proof ofa recent observation by Okuyama. As a by-product, we present a new calculationof the connected 2-point correlator of multiply-wound Wilson loops at leadingorder in $1/N$.

Note on generating functions and connected correlators of 1/2-BPS Wilson loops in $\mathcal{N}=4$ SYM theory