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On the Positive Moments of Ranks of Partitions

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