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 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 momentη 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 ofη 2k (n) and η 2k−1 (n), we show that for fixed k and i with 1 ≤ i ≤ k + 1,η 2k−1 (n) equals the number of (k + 1)-marked Durfee symbols of n with the i-th rank being zero andη 2k (n) equals the number of (k + 1)-marked Durfee symbols of n with the i-th rank being positive. The interpretations ofη 2k−1 (n) andη 2k (n) also imply the interpretation of η 2k (n) given by Andrews since η 2k (n) equalsη 2k−1 (n) plus twice ofη 2k (n). Moreover, we obtain the generating functions ofη 2k (n) andη 2k−1 (n).