  We consider two different physical systems for which the basis of the Hilbert
space can be parametrized by Young diagrams: free complex fermions and the
phase model of strongly correlated bosons. Both systems have natural,
well-known deformations parametrized by a parameter Q: the former one is
related to the deformed boson-fermion correspondence introduced by N. Jing,
while the latter is the so-called Q-boson, arising also in the context of
quantum groups. These deformations are equivalent and can be realized in the
same way in the algebra of Hall-Littlewood symmetric functions. Without a
deformation, these reduce to Schur functions, which can be used to construct a
generating function of plane partitions, reproducing a topological string
partition function on $C^3$. We show that a deformation of both systems leads
then to a deformed generating function, which reproduces topological string
partition function of the conifold, with the deformation parameter Q identified
with the size of $P^1$. Similarly, a deformation of the fermion one-point
function results in the A-brane partition function on the conifold.
