Weighted partition rank and crank moments. II: Odd-order moments
Let $\lambda$ be an integer partition. We write $\lambda\vdash n$ if $\lambda$ is a partition of $n$. Let $\sharp(\lambda)$, $\omega(\lambda)$, and $\ell(\lambda)$ be the number of parts in $\lambda$, the number of ones in $\lambda$, and the largest part in $\lambda$, respectively. The \textit{rank} of a partition $\lambda$ is defined by \textit{F. J. Dyson} [``Some guesses in the theory of partitions'', Eureka 8, 10--15 (1944)] as $$\text{rank}(\lambda):=\ell(\lambda)-\sharp(\lambda).$$ \textit{G. E. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167-171 (1988;Zbl 0646.10008)] defined the \textit{crank} of a partition $\lambda$ by $$ \text{crank}(\lambda):= \begin{cases} \ell(\lambda), & \text{ if }\omega(\lambda)=0,\\ \mu(\lambda)-\omega(\lambda), & \text{ if }\omega(\lambda)>0, \end{cases} $$ where $\mu(\lambda)$ denotes the number of parts in $\lambda$ larger than $\omega(\lambda)$. Let $N(m,n)$ (resp. $M(m, n)$) count the number of partitions of $n$ whose rank (resp. crank) is $m$. The $k$th ordinary rank and crank moments are, respectively, defined by $$ N_k(n):=\sum_{m=-\infty}^\infty m^k\, N(m,n) = \sum_{\lambda\vdash n} \text{rank}^k (\lambda) $$ and $$ M_k(n):=\sum_{m=-\infty}^\infty m^k\, M(m,n) = \sum_{\lambda\vdash n} \text{crank}^k (\lambda). $$ \textit{G. E. Andrews} [Invent. Math. 169, No. 1, 37-73 (2007;Zbl 1214.11116)] defined the $k$th symmetrized rank moment by $$ \eta_k(n) :=\sum_{m=-\infty}^\infty \binom{m+\lfloor \frac{k-1}{2}\rfloor}{k}\,N(m,n) = \sum_{\lambda\vdash n} \binom{\text{rank}(\lambda)+\lfloor \frac{k-1}{2}\rfloor}{k}. $$ \textit{F. G. Garvan} [Adv. Math. 228, No. 1, 241--265 (2011; Zbl 1268.11143)] defined the $k$th symmetrized crank moment by $$ \mu_k(n) :=\sum_{m=-\infty}^\infty \binom{m+\lfloor \frac{k-1}{2}\rfloor}{k}\,M(m,n) = \sum_{\lambda\vdash n} \binom{\text{crank}(\lambda)+\lfloor \frac{k-1}{2}\rfloor}{k}. $$ The $k$th weighted ordinary and symmetrized rank and crank moments are defined by the author as follows: \begin{align*} N_k^{\sharp}(n)&:=\sum_{\lambda\vdash n} \sharp(\lambda)\, \text{rank}^k (\lambda),\\ M_k^{\omega}(n)&:=\sum_{\lambda\vdash n} \omega(\lambda)\, \text{crank}^k (\lambda),\\ \eta_k^{\sharp}(n)&:=\sum_{\lambda\vdash n} \sharp(\lambda)\, \binom{\text{rank}(\lambda)+\lfloor \frac{k-1}{2}\rfloor}{k},\\ \mu_k^{\omega}(n)&:=\sum_{\lambda\vdash n} \omega(\lambda)\, \binom{\text{crank}(\lambda)+\lfloor \frac{k-1}{2}\rfloor}{k}. \end{align*} In this paper, the author proves that this weighted ordinary and symmetrized rank and crank moments of higher order are closely related to the corresponding rank and crank moments when the order is odd. For Part I see [the author, ``Weighted partition rank and crank moments. I: Andrews-Beck type congruences'', Conference Proceedings in Honor of Bruce Berndt (to appear)]
scientific article; zbMATH DE number 7472659