Asymptotic expression for raw moment of the multiplicity of a part in a general partition function. (English) Zbl 1530.05011
Summary: Partitions appear across physics like Bose-Einstein statistics, dimer coverings, Potts model, crystal growth, Vasiliev theory, quasicrystals, Penrose-like random tilings, and so on. Since the general analytical study of partitions is rather tough, so statistical study is immensely preferred. This paper derives an asymptotic formula to evaluate the \(m\)-th raw moment of the multiplicity of a part for a broad category of partitions given by a specific generating function. In addition to being a multifold refinement/generalization of results in the literature, this work allows to calculate useful statistical quantities like mean and variance of the number of appearance of a part which is of deeper significance to analysts. The obtained expression is numerically validated to be in close agreement with the actual values. This study instigates the development of a general paradigm where partition theory is exposed to more sophisticated statistical analysis, thus, opening new avenues of research in this field.
MSC:
| 05A17 | Combinatorial aspects of partitions of integers |
| 11P82 | Analytic theory of partitions |
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 82B03 | Foundations of equilibrium statistical mechanics |
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