Moments of ranks and cranks, and quotients of Eisenstein series and the Dedekind eta function (Q2079493)

The concepts of rank and crank of a partition of a positive integer was introduced by \textit{F. Dyson} [``Some guesses in the theory of partitions'', Eureka 8, 10--15 (1944)] in order to explain combinatorially the Ramanujan congruences \begin{align*} p(5n+4) &\equiv 0 \pmod 5,\\ p(7n+5) &\equiv 0 \pmod 7,\\ p(11n+6) &\equiv 0 \pmod {11}. \end{align*} \textit{A. O. L. Atkin} and \textit{F. G. Garvan} [Ramanujan J. 7, No. 1--3, 343--366 (2003; Zbl 1039.11069)] introduced the functions \(N_k(n)\) and \(M_k(n)\), which denote the \(k\)-th moments of ranks and cranks. In this paper, the authors investigate arithmetic properties of \(N_{2k}(n)\) and \(M_{2k}(n)\) by considering Fourier coefficients of some special weakly holomorphic modular forms of half-integral weights. Congruences for the moments and symmetrized moments of ranks and cranks were established in this context.