Ranks of Partitions and Durfee Symbols

Open Access
Author:
Keith, William Jonathan
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
June 27, 2007
Committee Members:
  • George E Andrews, Committee Chair
  • Wen Ching W Li, Committee Member
  • Ae Ja Yee, Committee Member
  • Martin Furer, Committee Member
Keywords:
  • partition theory
  • Durfee
Abstract:
This thesis presents generalizations of several partition identities related to the rank statistic. One set of these is new: $k$-marked Durfee symbols, as defined in a paper by Andrews. This paper extends and elaborates upon several congruence theorems presented in the paper that originated those objects, showing that an infinite family of such theorems exists. The number of $l$-marked Durfee symbols of $n$ are related to the distribution of ranks of partitions of $n$ modulo $2l+1$; the relationship is made explicit and explored in various directions. Another set of identities deals with the very classical theorem of Euler on partitions into odd and distinct parts. This was given bijective proof by Sylvester, giving occasion to discover new statistical equalities, which in turn were generalized to partitions into parts all $equiv c , (mod , m)$ by Pak, Postnikov, Zeng, and others. This work further extends the previous theorems to partitions with residues $(mod , m)$ that differ but do not change direction of difference, i.e. residues monotonically rise or fall. Attached as an appendix is a translation of the thesis of Dieter Stockhofe, emph{Bijektive Abbildungen auf der Menge der Partitionen einer naturlichen Zahl}. This is provided in support of the tools therefrom used in Chapter 4, as well as in the spirit of a service to the Anglophone mathematical community.