Weighted partition rank and crank moments. II: Odd-order moments
type: Item
instance of: scholarly article
title: Weighted partition rank and crank moments. II: Odd-order moments
zbMATH Open document ID: 1486.11127
author: Shane Chern
published in: The Ramanujan Journal
publication date: 2022-02-11T00:00:00Z
review text: 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 …
reviewed by: Mircea Merca
Mathematics Subject Classification ID: 11P84
Mathematics Subject Classification ID: 11P81
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 7472659
zbMATH Keywords: partition
zbMATH Keywords: rank
zbMATH Keywords: crank
zbMATH Keywords: weighted moment
MaRDI profile type: Publication
full work available at URL: s11139-020-00365-9
OpenAlex ID: W3130809731
cites work: Applications of Basic Hypergeometric Functions
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DOI: 10.1007/S11139-020-00365-9
description: scientific article; zbMATH DE number 7472659