  Using $\Gamma_{\pm}(z) $ vertex operators of the $c=1$ two dimensional
conformal field theory, we give a 2d-quantum field theoretical derivation of
the conjectured d- dimensional MacMahon function G$_{d}(q) $. We interpret this
function G$_{d}(q) $ as a $(d+1) $- point correlation function
$\mathcal{G}_{d+1}(z_{0},...,z_{d}) $ of some local vertex operators
$\mathcal{O}%_{j}(z_{j}) $. We determine these operators and show that they are
particular composites of q-deformed hierarchical vertex operators $% \Gamma
_{\pm}^{(p)}$, with a positive integer p. In agreement with literature's
results, we find that G$_{d}(q) $, $d\geq 4$, cannot be the generating
functional of all \textit{d- dimensional} generalized Young diagrams .
