Alladi Ramakrishnan Hall

Estimating the asymptotics of solid partitions

S. Govindarajan

Indian Institute of Technology, Madras, Chennai

We obtain 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 $n^(−3/4) \; log
p_3(n)$ tends to $1.822 \pm 0.001$ at large n. 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.