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arXiv · 2310.02796

Multimodal sequences and their generating functions

Philip Cuthbertson·Robert Schneider
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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.

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Philip Cuthbertson, Robert Schneider. 2023-10-04. Multimodal sequences and their generating functions. https://arxiv.org/abs/2310.02796

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