Compositions, Rook Placements, and Permutations of Vectors

ACKNOWLEDGEMENTS ....................................................................................... iii
CHAPTER
1 INTRODUCTION
1.1 Generating Functions and Partition Theory ........................... 1
1.2 Gaussian Polynomials ........................................................... 5
1.3 Ferrers Boards ................................................. 8
1.4 Permutation Statistics and Compositions ............................. 11
2 VECTOR VERSIONS OF Q-IDENTITIES
2.1 Inversions of Permutations and Partitions of Vectors . . . 14
2.2 The MACKOH Identity ............................... 36
3 COMPOSITIONS AND ROOK PLACEMENTS
3.1 Compositions of Vectors ..................................................... 48
3.2 Unitary Compositions ........................................................... 60
3.3 Simon Newcomb’s Problem .................................................. 66
4 COMPOSITIONS AND ROOK PLACEMENTS: Q-VERSIONS
4.1 q-Compositions ....................................................................... 84
4.2 q-Unitary Compositions ........................................................ 91
4.3 q-Rook Theory: Recursion Formulae .................................. 93
4.4 The (q-r) Simon Newcomb Problem and Unimodaliy. . . 110
3 CONCLUSION
3.1 Summary of Main Methods and Results ........................... 133
BIBLIOGRAPHY ................................................................................... 142
APPENDIX UST OF SYMBOLS