Estimating the Asymptotics of Solid Partitions

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→∞ n −3/4 log p 3 (n) ∼ 1.822±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.

Magnificent Four with Colors

“…where p 3 (k) counts the number of solid partitions of size k, does not have a known closed form; only its values up to k ∼ 72 and its asymptotic behavior for large k [9] are known. See Ref.…”

Stochastic dynamics of growing Young diagrams and their limit shapes

“…Let us start with the initial condition (7) and count the integrated current, i.e. the number of particles P t which enter the initially empty half-line x ≥ 0 at time t. Since the density is asymptotically n = 1 2 at the origin, see (15), the current is n(1 − n) = 1/4 and the average integrated current is t/4. The more precise behavior is [56]…”

MacMahon's statistics on higher-dimensional partitions

“…It was conjectured in [BGP12] and (for solid partitions) in [MR03] supported by numerical experiments, that p d (n), the number of d-dimensional partitions of volume (size) n, has exactly the same asymptotics. However, later computations reported in [DG15] suggest that this is not the case (for d = 3) and that p 3 (n) is asymptotically larger than m 3 (n) (despite the fact that m 3 (n) = p 3 (n) for n ≤ 5 and m 3 (n) > p 3 (n) for the next many values of n [ABMM67, DG15]; cf. the sequences A000293, A000294 in [OEIS]).…”

“…Despite long interest and many connections to various fields including algebra, combinatorics, geometry, probability and statistical physics, the subject remains rather mysterious-very little is known about d-dimensional partitions for d ≥ 3. See [ABMM67,Knu70,Gov13] on some computational and enumerative aspects; [MR03,BGP12,DG15] on asymptotic data and connections to physics; [BBS13,Nek17,CK18] on further aspects particularly related to the theory of Donaldson-Thomas invariants. (See also the remarks and references in final Sec.…”