Identities and inequalities for the \(M_2\)-rank of partitions without repeated odd parts modulo 8 (Q2155876)

In order to combinatorially explain the famous Ramanujan congruences \[ p(5n+4)\equiv 0 \bmod 5, \quad p(7n+5)\equiv 0 \bmod 7,\text{ and } p(11n+6)\equiv 0 \bmod 11 \] for the number \(p(n)\) of the partitions of \(n,\) \textit{F. J. Dyson} [Eureka 8, 10--15 (1944)] defined the rank of a partition as the largest part minus the number of parts. He conjectured that this statistic could be utilized to show the mod 5 and mod 7 congruences, which was later confirmed by \textit{A. O. L. Atkin} and \textit{H. P. F. Swinnerton-Dyer} [Proc. London Math. Soc. 4, 84--106 (1954; Zbl 0055.03805)]. \textit{A. Berkovich} and \textit{F. G. Garvan} [J. Comb. Theory, Ser. A 100, 61--93 (2002; Zbl 1016.05003)] defined the \(M_2\)-rank of a partition \(\lambda\) without repeated odd parts as \(\lceil\ell(\lambda)/2\rceil-\nu(\lambda)\) where \(\ell(\lambda)\) is the largest part of \(\lambda\) and \(\nu(\lambda)\) is the number of parts of \(\lambda\). Let \(N_2(a, M, n)\) denote the number of partitions of \(n\) without repeated odd parts whose \(M_2\)-rank is congruent to \(a\) modulo \(M\). \textit{J. Lovejoy} and \textit{R. Osburn} [J. Théor. Nombres Bordeaux 21, 313--334 (2009; 1270.11103)] and \textit{R. Mao} [Ramanujan J. 37, 391--419 (2015; Zbl 1328.11104)] proved a number of results for \(M_2\)-rank differences modulo 3, 5, 6, and 10. In the paper under review the authors establish the generating functions for \(N_2(a, 8, n)\) with \(0\le a \le 7\) using Appell-Lerch sums. They prove some equalities and inequalities on \(M_2\)-rank modulo 8 of partitions without repeated odd parts, e.g., for \(n\ge 0,\) \[N_2(2, 8, 4n+2)=N_2(1, 8, 4n+2), \quad N_2(2, 8, 4n+4)<N_2(1, 8, 4n+4).\] The authors also relate some \(M_2\)-rank differences with eighth-order mock theta funcions \(T_0(q)\) and \(T_1(q)\) due to \textit{B. Gordon} and \textit{R. J. McIntosh} [J. London Math. Soc. 62, 321--335 (2000; Zbl 1031.11007)].