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 $1\leq i\leq k+1$, $\bar{ \eta}_{ 2k-1}(n)$ 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 $\bar{ \eta}_{ 2k-1}(n)$.
Categories: math.CO math.NT