Hall–Littlewood Plane Partitions and KP

MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletić to a generating function of weighted plane partitions that is related to Hall–Littlewood polynomials, $\mathcal S(t)$, and further to one related to Macdonald polynomials, $\mathcal S(t,q)$. Using Jing's one-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall–Littlewood extension. Confining the plane partitions to a finite $s\times s$ square base, we show that the resulting generating function, $\mathcal S_{s\times s}(t)$, is an evaluation of a $\tau$-function of KP.