The author begins the first half of the paper by defining the symmetrized \(k\)th rank moment, \[ \eta_k(n) = \sum_{m \in \mathbb{Z}} \begin{pmatrix} m + \lfloor \frac{k-1}{2} \rfloor \\ k \end{pmatrix} N(m,n), \] where \(N(m,n)\) is the number of partitions of \(n\) with rank \(m\), and computing its generating function, \[ \sum_{n \geq 1} \eta_{2v}(n)q^n = \prod_{n \geq 1} \frac{1}{1-q^n} \sum_{n \in \mathbb{Z} \backslash \{0\}} \frac{(-1)^{n-1}q^{n(3n-1)/2 + vn}}{(1-q^n)^{2v}}. \] (Odd rank moments are identically \(0\).)
He then defines and computes the generating function for \(k\)-marked Durfee symbols arising from partitions of \(n\). If \(\mathcal{D}_k(n)\) denotes the number of such symbols, then \(\mathcal{D}_1(n) = p(n)\), the number of partitions of \(n\), and for \(k \geq 1\) we have \(\mathcal{D}_{k+1}(n) = \eta_{2k}(n)\).
Next the author discusses the notions of rank and conjugation for \(k\)-marked Durfee symbols. He defines the full rank, which generalizes Dyson’s rank of a partition and provides combinatorial interpretations of certain congruences. For instance, one has \(\mathcal{D}_2(n) \equiv 0 \pmod{5}\) if \(n \equiv \pm 1 \pmod{5}\), which follows from the identity \(NF_2(r,5,5n+a) = \frac{1}{5}\mathcal{D}_2(5n+a)\), where \(a=1\) or \(4\) and \(NF_2(r,t,n)\) is the number of \(2\)-marked Durfee symbols of \(n\) whose full rank is congruent to \(r\) modulo \(t\).
In the second half of the paper the author considers \(k\)-marked odd Durfee symbols and moments of odd ranks. When \(k=1\) the generating function for odd Durfee symbols is the third order mock theta function \(\omega(q)\).
The paper ends with a discussion of multiple series representations of mock theta functions and a list of suggested research problems.