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 Baileys 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 ${pover 3}<p< {pover 2}$ 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.
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