Multimodal sequences and their generating functions

Abstract

We define integer multimodal sequences, which are generalizations of unimodal sequences having multiple local peaks of equal size. The generating functions for multimodal sequences represent novel types of $q$-series that combine generating functions for both integer partitions and integer compositions. We prove a bijection between multimodal sequences of equal size (sum), and show that multimodal generating functions become finite series at roots of unity like the ``strange'' function of Kontsevich, quantum modular forms, and other examples of this phenomenon in the $q$-series literature.