Asymptotic Formulae for Restricted Unimodal Sequences

Abstract

Additive enumeration problems, such as counting the number of integer partitions, lie at the intersection of various branches of mathematics including combinatorics, number theory, and analysis. Extending partitions to integer unimodal sequences has also yielded interesting combinatorial results and asymptotic formulae, which form the subject of this thesis. Much like the important work of Hardy and Ramanujan proving the asymptotic formula for the partition function, Auluck and Wright gave similar formulas for unimodal sequences. Following the circle method of Wright, we provide the asymptotic expansion for unimodal sequences with odd parts. This is then generalized to a two-parameter family of mixed congruence relations, with parts on one side with parts on one side up to the peak satisfying r (mod m) and parts on the other side -r (mod m), and an asymptotic formula is provided. Techniques used in the proofs include Wright's circle method, modular transformations, and bounding of complex integrals.