  In this work, we give combinatorial proofs for generating functions of two
problems, i.e., flushed partitions and concave compositions of even length. We
also give combinatorial interpretation of one problem posed by Sylvester
involving flushed partitions and then prove it. For these purposes, we first
describe an involution and use it to prove core identities. Using this
involution with modifications, we prove several problems of different nature,
including Andrews' partition identities involving initial repetitions and
partition theoretical interpretations of three mock theta functions of third
order $f(q)$, $\phi(q)$ and $\psi(q)$. An identity of Ramanujan is proved
combinatorially. Several new identities are also established.
