Sets of rigged paths with Virasoro characters

Let \{M_{r,s}\}_{0< r < p, 0< s < p'} be the irreducible Virasoro modules in the $(p,p')$-minimal series. In our previous paper, we have constructed a monomial basis of \oplus_{r=1}^{p-1}M_{r,s} in the case of $1<p'/p<2$. By `monomials' we mean vectors of the form \phi^{(r_L,r_{L-1})}_{-n_L}...\phi^{(r_1,r_{0})}_{-n_1} |r_0,s >, where \phi_{-n}^{(r',r)} are the Fourier components of the (2,1)-primary field mapping M_{r,s} to M_{r',s}, and |r_0,s > is the highest weight vector of M_{r_0,s}. In this article, for all p<p' with p>2 and s=1, we describe a subset of such monomials which conjecturally forms a basis of \oplus_{r=1}^{p-1}M_{r,1}. We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case of p=3.