Contents 1. Prologue 1 1.1. Introduction 1 1.2. Structure of this paper 5 1.3. Continued fractions 6 1.4. Fermionic character expressions 6 1.5. The Takahashi and string lengths 8 1.6. Takahashi trees 8 1.7. Takahashi tree vectors 10 1.8. Truncated Takahashi tree 11 1.9. The linear term 12 1.10. The constant term 12 1.11. The quadratic term 13 1.12. The ran-system and the parity vector 14 1.13. The extra term 14 1.14. Finitized fermionic expressions 15 1.15. Fermionic-like expressions 17 2. Path combinatorics 19 2.1. Outline of proof 19 2.2. The band structure 21 2.3. Weighting function 22 2.4. Winged generating functions 24 2.5. Striking sequence of a path 25 2.6. Path parameters 26 2.7. Scoring generating functions 27 2.8. A seed 28 2.9. Partitions 28 3. The S-transform 29 3.1. Path-dilation: the Bi-transform 29 3.2. Particle insertion: the S2-transform 34 3.3. Particle moves 35 3.4. The ^-transform 37 3.5. Particle content of a path 38 4. The P-transform 41 4.1. The pure P-transform 41 4.2. The SP-pair 42 V
Previous Page Next Page