  The generating function of partitions with repeated (resp. distinct) parts
such that each odd part is less than twice the smallest part is shown to be the
third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results
for partitions with the corresponding restriction on each even part are also
obtained, one of which involves the third order mock theta function $\phi(q)$.
Congruences for the smallest parts functions associated to such partitions are
obtained. Two analogues of the partition-theoretic interpretation of Euler's
pentagonal theorem are also obtained.
