Bijective proofs of partition identities and covering systems

partitions;bijective proofs;covering systems

Kim, Sun

关键词: partitions;    bijective proofs;    covering systems;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/15597/1_Kim_Sun.pdf?sequence=2&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This dissertation involves two topics. The first is on the theory of partitions, which is discussed inChapters 2 − 5. The second is on covering systems, which are considered in Chapters 6 − 8.In 2000, Farkas and Kra used their theory of theta functions to establish a beautiful theorem oncolored partitions, and they asked for a bijective proof of it. In Chapter 2, we give a bijective proofof a more general partition identity, with the Farkas and Kra partition theorem being a special case.We then derive three further general partition identities and give bijective proofs of these as well.The quintuple product identity is one of the most famous and useful identities in the theoryof theta functions and q-series, and dates back to 1916 or earlier. In his recent survey paper onthis identity, Shaun Cooper remarked that there does not exist a bijective proof of it. In Chapter3, employing bijective proofs of Jacobi’s triple product identity and Euler’s pentagonal numbertheorem, we provide the first bijective proof of the quintuple product identity.In a recent paper, The parity in partition identities, George Andrews investigated parity questionsin partition identities and listed 15 open problems at the end of his paper. In Chapter 4, weprovide solutions to the first two open problems suggested by Andrews. More precisely, we providecombinatorial proofs of two partition identities which were derived by comparing Andrews’new identity with G¨ollnitz-Gordon identities or certain generalizations thereof.In our last chapter on partitions, Chapter 5, we give a combinatorial proof of a companionto Euler’s famous recurrence formula for the sum of divisors function. Euler’s recurrenceformula had previously been combinatorially proved using a double counting argument, but itsequally famous companion has not heretofore been established combinatorially. We not only providesuch a combinatorial proof, but we also give a combinatorial proof of a vast generalization as well.M. Filaseta, K. Ford, S. Konyagin, C. Pomerance and G. Yu proved that if the least modulusN of a covering system is sufficiently large, then the sum of reciprocals of the moduli is boundedbelow by a function of N, tending to infinity as N goes to infinity, which confirms a conjecture of P. Erd˝os andJ. L. Selfridge. They also showed that, forK > 1, the complement in Z of any union of residueclasses r(n) (mod n) with distinct n from (N,KN] has density at least dK for N sufficiently large,which implies a conjecture of P. Erd˝os and R. L. Graham. In Chapter 6, we first define coveringsystems in number fields, and extend those results to arbitrary number fields.In Chapter 7, we give an explicit version of their first theorem to provide a specific number forthe least modulus of a covering system, where the reciprocal sum is strictly bigger than 1.In the last chapter, Chapter 8, we consider exact covering systems in number fields. Motivatedby the theorem of Davenport, Mirsky, Newman and Rado that there does not exist an exact coveringsystem with distinct moduli, we raise the question whether or not this is true for covering systemsin algebraic number fields. We provide affirmative answers for certain quadratic fields.