An analogue of $k$-marked Durfee symbols for strongly unimodal sequences

TL;DR

This paper introduces $k$-marked strongly unimodal symbols, a new combinatorial class generalizing strongly unimodal sequences, and explores their rank generating functions and potential quantum modularity properties.

Contribution

It defines $k$-marked strongly unimodal symbols, extending Andrews' $k$-marked Durfee symbols, and studies their multivariate rank generating functions and modularity aspects.

Findings

Established a multivariate rank generating function for the new objects.

Connected the generating functions to potential quantum modularity properties.

Provided combinatorial interpretations and properties of the new symbols.

Abstract

In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called $k$-marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting modularity properties at certain vectors of roots of unity. Motivated by recent studies of rank generating functions for strongly unimodal sequences, we apply methods of Andrews to define an analogous class of combinatorial objects called $k$-marked strongly unimodal symbols that generalize strongly unimodal sequences. We establish a multivariate rank generating function for these objects, which we study combinatorially. We conclude by discussing potential quantum modularity properties for this rank generating function at certain vectors of roots of unity.