Odd Coefficients of Weakly Holomorphic Modular Forms

s 1. Fisseha Abebe, Clark Atlanta University Distance Metric for Protein Partitions The composition of proteins in terms of amino-acid sequences within three protein theoretical families: serine, leucine, and valine, with respective sizes of 8, 7, and 5 amino acids, have been studied using combinatorial methods. All possible partitions are examined, and a metric is defined to measure the distance between any random partition and the natural partition of the genetic code. Distribution of distance measures has shown patterns of departures from the natural partition. Joint work with William Seffens, Clark Atlanta University. 2. George E Andrews, Pennsylvania State University Multipartition Identities and the Tri-Pentagonal Number Therem In the Journal of Combinatorial Theory, 91(2000), 464-475, Bailey chains were extended to multiple series, and new Pentagonal Number Theorems were deduced. The triple series involving pentagonal numbers (= Tri-Pentagonal Number Theorem) was most intriguing. In the first half of this talk, we shall both interpret the related triple q-series as the generating function for certain tri-partitions, and we shall show that the triple pentagonal number side of the identity can be reduced to a linear combination of three infinite products. In the last half, we shall then discuss the further possibilities of multipartition identities and congruences. In these latter considerations, modular forms, mock theta functions, false theta functions and total interlopers make mysterious appearances. 3. Mohamed El Bachraoui, United Arab Emirates University Mobius Inversion Formulas for Functions of Several Variables We extend the Mobius inversion formula for functions of one single variable to functions of several variables. As applications, we count for n ≥ m the number of relatively prime subsets of sets of the form {m,m + 1, . . . , n} and we give a fomrula for phi functions for such sets. Our results generalize the work of M. Nathanson on relatively prime sets and phi functions for sets {1, 2, . . . , n}.