# Estimating the Asymptotics of Solid Partitions

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## Abstract

We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If \(p_3(n)\) denotes the number of solid partitions of an integer \(n\), we show that \(\lim _{n\rightarrow \infty } n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001\). This shows clear deviation from the value \(1.7898\), attained by MacMahon numbers \(m_3(n)\), that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in \(\log p_3(n)\). In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to \(n^{1/4}\), the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.

## Keywords

Solid partitions of an integer Asymptotic expansion Transition matrix Monte Carlo simulations## Notes

### Acknowledgments

SG would like to thank Intel India for financially supporting the numerical study of solid partitions. We also thank the High Performance Computing Environment at IIT Madras which provided us access to the Virgo and Vega super clusters where our Monte Carlo simulations were carried out.

## Supplementary material

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