  We derive the fermionic polynomial generalizations of the characters of the
integrable perturbations $\phi_{2,1}$ and $\phi_{1,5}$ of the general minimal
$M(p,p')$ conformal field theory by use of the recently discovered trinomial
analogue of Bailey's lemma. For $\phi_{2,1}$ perturbations results are given
for all models with $2p>p'$ and for $\phi_{1,5}$ perturbations results for all
models with ${p'\over 3}<p< {p'\over 2}$ are obtained. For the $\phi_{2,1}$
perturbation of the unitary case $M(p,p+1)$ we use the incidence matrix
obtained from these character polynomials to conjecture a set of TBA equations.
We also find that for $\phi_{1,5}$ with $2<p'/p < 5/2$ and for $\phi_{2,1}$
satisfying $3p<2p'$ there are usually several different fermionic polynomials
which lead to the identical bosonic polynomial. We interpret this to mean that
in these cases the specification of the perturbing field is not sufficient to
define the theory and that an independent statement of the choice of the proper
vacuum must be made.
