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

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.