ASYMPTOTICS FOR MOMENTS OF HIGHER RANKS

The odd moments of ranks and cranks

have motivated much research. Here, p(n) denotes the number of partitions of n. In particular, toward a combinatorial explanation of the above congruences many partition statistics have been studied. Among them, the rank suggested by F. Dyson [6] and the crank suggested by the first author and F.G. Garvan [2] have proven successful and their own properties have been extensively studied. Here, t...

Asymptotics for Rank and Crank Moments

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to two of the authors that re ned a conjecture of Garvan. Garvan's original conjecture states that the moments of the crank function are always larger than the mome...

Convexity Ranks in Higher Dimension

The Method of Moments for Higher Moments { Problems , Solutions , and

Though often not explicitly stated, the method of moments is one of the heavily used methods in queueing analysis. A typical, tagged customer is traced as it proceeds through the system and the components of its delay are individually analyzed. Using the moments of these components, moments of the sojourn and waiting times of an arbitrary customer can be derived. By this method, expected values...

On the Positive Moments of Ranks of Partitions

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 η̄k(n) of ranks of partitions of n which is defined as the truncated sum over positive ranks of partitions of n. As combintorial interpret...