On new representations for partitions

• Grant number: 12/01258-9
• Support Opportunities: Scholarships abroad - Research
• Start date: July 10, 2012
• End date: December 19, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: José Plínio de Oliveira Santos
• Grantee: José Plínio de Oliveira Santos
• Host Investigator: George E. Andrews
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Institution abroad: Pennsylvania State University, United States

Abstract

By the introduction of parameters in identities of the Rogers-Ramanujan type we were able to obtain some functional equations and by studying these equations to get combinatorial interpretations for partitions. Later we found that it was more convenient to write them as two-line arrays. First we found three distinct interpretations for unrestricted partitions with an explicit bijection between the partitions and the corresponding matrices for just one of those. Latter we were able to get the bijections for the other two. By using these two-line representations it was possible to give a new combinatorial proof for an interesting result given by Andrews related to three quadrant Ferrers graphs. It is important to mention that we know how to get the two-line arrays in two different ways. An important property of these representations is the fact that one can get q-analogs for unrestricted partitions and all the ones having the two-line representation. One way to do this is by considering the area under the path on the first quadrant that can be constructed using those matrices. These are similar to what has been done by Polya to give a combinatorial interpretation for the Gaussian polynomials that are q-analog for the binomial numbers. I have to mention that we were able to find, by our method, combinatorial interpretations for all the Mock Theta Functions as two-line matrices. We are planning to continue working in this direction, in questions of combinatorial nature related to identities of the Rogers-Ramanujan type and, also, in other questions of enumerative combinatorics. In this mater I mention the enumeration of (0,1)-symmetric matrices with a given sum for the entries in each row using generation functions. This result is part of a paper in preparation. (AU)