On the Positive Moments of Ranks of Partitions
William Y.C. Chen, Kathy Q. Ji, and Erin Y.Y. Shen
Abstract: By introducing k-marked Durfee symbols, Andrews found a combinatorial interpretation of 2k-th symmetrized moment η2k(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 of ranks of partitions of n which is defined as the truncated sum over positive ranks of partitions of n. As combintorial interpretations of and , we show that for given k and i with 1 ≤ i ≤ k + 1, equals the number of (k + 1)-marked Durfee symbols of n with the i-th rank being zero and equals the number of (k + 1)-marked Durfee symbols of n with the i-th rank being positive. The interpretations of and are independent of i, and they imply the interpretation of η2k(n) given by Andrews since η2k(n) equals plus twice of . Moreover, we obtain the generating functions of and .
AMS Classification: 05A17, 11P83, 05A30.
Keywords: rank of a partition, k-marked Durfee symbol, moment of ranks