| The common thread in the five chapters is that partitions of integers play at least a marginal role in them. Three can be classified as belonging to number theory, more precisely partitions and basic hypergeometric series, one of them as belonging to combinatorics and one to linear algebra using some tools from approximation theory.; Problem one. Every partition of an integer n can be reduced to its unique t-core, where t is a fixed positive integer. The number of t-cores of n is related to an expression of {dollar}eta{dollar}-functions. There are several ways (developed by F. Garvan, D. Stanton and others) to calculate this number. By introducing a new coordinate system, a simpler method is obtained.; A q-series identity of N. Fine's shows that the number of 3-core pair solutions {dollar}(x,y){dollar} of {dollar}vert xvert + 2cdot vert yvert = 3n + 2{dollar} is three times the number of solutions of {dollar}vert xvert + 2cdot vert yvert = n.{dollar} Surprisingly, two combinatorial statistics are found to explain this fact. The solutions are classified using both of them. This leads to a combinatorial explanation of a congruence of P. C. Eggans and an interesting self-similarity, among others.; There is also a new, constructive proof of the old fact that the number of t-cores of {dollar}tn + t - {lcub}tsp2-1over 24{rcub}{dollar} can be divided by t, for t = 5, 7, 11.; Problem two. As a complement to Dyson's rank, the notion of frame is introduced by the author to denote the (length of the largest part) + (number of parts) {dollar}-{dollar}1. Various properties of the numbers {dollar}psb{lcub}r{rcub}(n){dollar} (the number of partitions of n with frame size r) are noted, including a recursion formula. If the frame is even, then {dollar}psb{lcub}r{rcub}(n){dollar} is even, and this might help determine the parity of the number of partitions of n, {dollar}p(n),{dollar} since {dollar}sumsb{lcub}r{rcub} psb{lcub}r{rcub}(n)= p(n).{dollar} The sum {dollar}sumsb{lcub}n{rcub} psb{lcub}r{rcub}(n)= 2sp{lcub}r-1{rcub}{dollar} is also obtained.; Problem three. The classical Bailey transform and the Bailey lemma contain a surprisingly large slice of the theory of the basic hypergeometric series. G. Andrews iterated constant matrices to gain more complex identities and later, with A. Agarwal and D. Bressoud, was able to introduce some change in the matrices. The generalization here can be regarded as a careful analysis and rethinking of their approach.; Problem four. Affine transformations, used in Problem One, are also important in several other fields such as fractal theory (image compression), robotics, etc. Recurrence sequences are further examples of affine transformations. Using some approximation theory, an algorithm and formulas are developed for iterated affine transformations, the results are decomposed for quicker computation in array computers.; Problem five. The handling of convolutions of sequences is unified. This naturally leads to determinants of matrices whose entries above the superdiagonal are all zeros. By analyzing their structure, it is shown how these determinants specialize to produce the major combinatorial numbers, such as the binomial coefficients, Stirling numbers of either kind. (Abstract shortened by UMI.)... |
