Memoirs of the American Mathematical Society 2005; 160 pp; softcover Volume: 175 ISBN10: 0821836560 ISBN13: 9780821836569 List Price: US$67 Individual Members: US$40.20 Institutional Members: US$53.60 Order Code: MEMO/175/827
 Fermionic expressions for all minimal model Virasoro characters \(\chi^{p, p'}_{r, s}\) are stated and proved. Each such expression is a sum of terms of fundamental fermionic form type. In most cases, all these terms are written down using certain trees which are constructed for \(s\) and \(r\) from the Takahashi lengths and truncated Takahashi lengths associated with the continued fraction of \(p'/p\). In the remaining cases, in addition to such terms, the fermionic expression for \(\chi^{p, p'}_{r, s}\) contains a different character \(\chi^{\hat p, \hat p'}_{\hat r,\hat s}\), and is thus recursive in nature. Bosonicfermionic \(q\)series identities for all characters \(\chi^{p, p'}_{r, s}\) result from equating these fermionic expressions with known bosonic expressions. In the cases for which \(p=2r\), \(p=3r\), \(p'=2s\) or \(p'=3s\), RogersRamanujan type identities result from equating these fermionic expressions with known product expressions for \(\chi^{p, p'}_{r, s}\). The fermionic expressions are proved by first obtaining fermionic expressions for the generating functions \(\chi^{p, p'}_{a, b, c}(L)\) of length \(L\) ForresterBaxter paths, using various combinatorial transforms. In the \(L\to\infty\) limit, the fermionic expressions for \(\chi^{p, p'}_{r, s}\) emerge after mapping between the trees that are constructed for \(b\) and \(r\) from the Takahashi and truncated Takahashi lengths respectively. Table of Contents  Prologue
 Path combinatorics
 The \(\mathcal{B}\)transform
 The \(\mathcal{D}\)transform
 Mazy runs
 Extending and truncating paths
 Generating the fermionic expressions
 Collating the runs
 Fermionic character expressions
 Discussion
 Appendix A. Examples
 Appendix B. Obtaining the bosonic generating function
 Appendix C. Bands and the floor function
 Appendix D. Bands on the move
 Appendix E. Combinatorics of the Takahashi lengths
 Bibliography
