Path generating transforms
  We study combinatorial aspects of q-weighted, length-L Forrester-Baxter paths, P^{p, p'}_{a, b, c}(L), where p, p', a, b, c \in Z_{+}, 0 < p < p', 0 < a, b, c < p', c = b \pm 1, L+a-b \equiv 0 (mod 2), and p and p' are co-prime.   We obtain a bijection between P^{p, p'}_{a, b, c}(L) and partitions with certain prescribed hook differences. Thereby, we obtain a new description of the q-weights of P^{p, p'}_{a, b, c}(L).   Using the new weights, and defining s_0 and r_0 to be the smallest non-negative integers for which |p s_0 - p' r_0|=1, we restrict the discussion to P^{p, p'}_{s_0} \equiv P^{p, p'}_{s_0,s_0,s_0+1}(L), and introduce two combinatorial transforms:   1. A Bailey-type transform B:   P^{p, p'}_{s_0}(L) -> P^{p, p'+p}_{s_0 + r_0}(L'), L \leq L',   2. A duality-type transform D:   P^{p, p'}_{s_0}(L) -> P^{p'-p, p'}_{s_0}(L).   We study the action of B and D, as q-polynomial transforms on the P^{p, p'}_{s_0}(L) generating functions, \chi^{p, p'}_{s_0}(L). In the limit L -> \infinity, \chi^{p, p'}_{s_0}(L) reduces to the Virasoro characters, \chi^{p, p'}_{r_0, s_0}, of minimal conformal field theories M^{p, p'}, or equivalently, to the one-point functions of regime-III Forrester-Baxter models.   As an application of the B and D transforms, we re-derive the constant-sign expressions for \chi^{p, p'}_{r_0, s_0}, first derived by Berkovich and McCoy. 
Omar Foda (Melbourne), Keith S. M. Lee (Melbourne), Yaruslav Pugai   (Melbourne, Landau), Trevor A. Welsh (Melbourne)