Rank deviations for overpartitions (Q6590162)

Recent investigations reveal unexpected biases for the number of partitions with parts in specified congruence classes. This paper investigates overpartitions, that is, partitions in which the first occurrence of each distinct part may be overlined, and their ranks. There are two ranks of interest. The (ordinary) rank of a partition \(\pi\) is the largest part \(\ell(\pi)\) minus the number of parts \(n(\pi)\). The \(M_2\)-rank is \(\lceil\ell(\pi)/2\rceil-n(\pi)+n(\pi_o)-\chi(\pi)\), where \(\pi_o\) is the subpartition consisting of the odd nonoverlined parts and \(\chi(\pi)=1\) if the largest part of \(\pi\) is odd and nonoverlined and 0 otherwise. The overparition rank deviations are \N\[\N\overline{D}(a,M)=\sum_{n\ge 0}\bigg(\overline{N}(a,M,n)-\frac{\overline{p}(n)}{M}\bigg)q^n \quad\mbox{and}\quad\overline{D}_2(a,M)=\sum_{n\ge 0}\bigg(\overline{N}_2(a,M,n)-\frac{\overline{p}(n)}{M}\bigg)q^n,\N\]\Nwhere \(\overline{N}(a,M,n)\) denotes the number of overpartitions of \(n\) with rank congruent to \(a\) modulo \(M\), \(\overline{N}_2(a,M,n)\) denotes the number of overpartitions of \(n\) with \(M_2\)-rank congruent to \(a\) modulo \(M\) and \(\overline{p}\) is the number of overpartitions of \(n\). The main results give explicit computations for pairs of rank deviations \(\overline{D}(a,M)+\overline{D}(a-1,M)\) and \(\overline{D}_2(a,M)+\overline{D}_2(a-1,M)\) in terms of Appell-Lerch series and sums of quotients of theta series. It is possible to extract results for a single rank deviations using the symmetry \(\overline{D}(a,M)=\overline{D}(M-a,M)\) and a similar result for \(\overline{D}_2\).\N\NThe approach is based on a method developed by [\textit{D. Hickerson} and \textit{E. Mortenson}, Math. Ann. 367, No. 1--2, 373--395 (2017; Zbl 1367.11077)] to study the rank deviation for ordinary partitions.\N\NThe cases \(M=3\) and \(M=6\) are simply stated in terms of \N\[\Nh(x;q) = \frac{(-q)_\infty}{(q)_\infty} \sum_{n\in\mathbb{Z}} \frac{(-1)^nq^{n^2+n}}{1-xq^n} \quad\mbox{and}\quad J_m=(q^m;q^m)_\infty.\N\]\NWe have \(\overline{D}(3,3)+\overline{D}(2,3) = -2q^2h(q^6;q^9)+J_2J_3^6J_{18}/3J_1^2J_6^3J_9^2\) and \(\overline{D}(2,3)+\overline{D}(1,3)=2\overline{D}(2,3) = 4q^2h(q^6;q^9)-2J_2J_3^6J_{18}/3J_1^2J_6^3J_9^2\).