Multimodal sequences and their generating functions

TL;DR

This paper introduces integer multimodal sequences with multiple peaks, explores their generating functions as novel $q$-series, and reveals their connection to quantum modular forms and finite series at roots of unity.

Contribution

It defines multimodal sequences, establishes a bijection between sequences of equal size, and analyzes their generating functions' special properties at roots of unity.

Findings

Multimodal sequences generalize unimodal sequences with multiple peaks.

Generating functions for these sequences are novel $q$-series.

They become finite series at roots of unity, similar to quantum modular forms.

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.