On the positive moments of ranks of partitions

William Y. C. Chen ; Kathy Q. Ji ; Erin Y. Y. Shen

The electronic journal of combinatorics, Tome 21 (2014) no. 1

Résumé

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of the $2k$-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 combinatorial interpretations of $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$, we show that for given $k$ 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 $\bar{\eta}_{2k}(n)$ are independent of $i$, and they 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 for $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$.

Détail

DOI : 10.37236/3852

Classification : 05A17, 05A15, 11P83, 05A30, 11P81
Mots-clés : rank of a partition, \(k\)-marked Durfee symbol, moment of ranks