Parity Considerations in Andrews-Gordon Identities and the k-Marked Durfee Symbols

Open Access
Author:
Kursungoz, Kagan
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
January 13, 2009
Committee Members:
  • George E Andrews, Dissertation Advisor
  • George E Andrews, Committee Chair
  • Ae Ja Yee, Committee Member
  • Woodrow Dale Brownawell, Committee Member
  • G Jogesh Babu, Committee Member
Keywords:
  • cluster parity index
  • parity index
  • Gordon marking
  • integer partition
  • k-marked Durfee symbol
Abstract:
In 1974, Andrews discovered the generating function for the partitions of n considered in a theorem due to Gordon. In a more recent paper, he reconsidered this generating function and gave refinements where additional restrictions involving parities are imposed. A combinatorial construction for the partitions enumerated by the mentioned generating function is given, and some of the Andrews' refinements are proven combinatorially. Some open problems posed by Andrews are solved, and a conjecture is settled. In another recent paper, Andrews defined k-marked Durfee symbols, and gave a substantial analytic account. He left the combinatorial explanations as open problems. Some of those open problems are also solved.