On the Positive Moments of Ranks of Partitions

Abstract

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment ${\eta }_{2k}(n)$ of ranks of partitions of $n$ in terms of $(k+1)$-marked Durfee symbols of $n$. In this paper, we consider the $k$-th symmetrized positive moment ${\bar{\eta }}_k(n)$ of ranks of partitions of $n$ which is defined as the truncated sum over positive ranks of partitions of $n$. As combintorial interpretations of ${\bar{\eta }}_{2k}(n)$ and ${\bar{\eta }}_{2k-1}(n)$, we show that for fixed $k$ and $i$ with $1\le i\le k+1$, ${\bar{\eta }}_{2k-1}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being zero and ${\bar{\eta }}_{2k}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being positive. The interpretations of ${\bar{\eta }}_{2k-1}(n)$ and ${\bar{\eta }}_{2k}(n)$ also imply the interpretation of ${\eta }_{2k}(n)$ given by Andrews since ${\eta }_{2k}(n)$ equals ${\bar{\eta }}_{2k-1}(n)$ plus twice of ${\bar{\eta }}_{2k}(n)$. Moreover, we obtain the generating functions of ${\bar{\eta }}_{2k}(n)$ and ${\bar{\eta }}_{2k-1}(n)$.

arXiv comment10 pages