Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals

Bhatia, D. P.; Prasad, M. A.; Arora, D.

doi:10.1088/0305-4470/30/7/010

Cited by 18 publications

(21 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The number of partitions of an integer (see [1,2] for an introduction) is one such enumeration problem with a history dating back to Euler. Examples of applications to physical problems include the q → ∞ Potts model [3], compact lattice animals [3,4], crystal growth [5], lattice polygons [6], BoseEinstein statistics [7,8] and dimer coverings [9]. The solution to the integer partitioning problem is known for 1-dimensional and 2-dimensional partitions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical estimation of the asymptotic behaviour of solid partitions of an integer

Mustonen

Rajesh

2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang-Landau sampling method for integers up to 8000. It is shown that limn→∞ ln(p 3 (n)) n 3/4 = 1.79 ± 0.01, where p3(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.

show abstract

Section: Introductionmentioning

confidence: 99%

“…[12]. While the generating functions for three and higher dimensional partitions are not known, it is known that lim n→∞ ln (p d (n)) /n d/(d+1) has a finite non-zero limit [4]. We define α d to be…”

Section: Introductionmentioning

confidence: 99%

Numerical estimation of the asymptotic behaviour of solid partitions of an integer

Mustonen

Rajesh

2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

show abstract

“…Let α > 0 be a fixed (small) constant. There are constant c 1 and c 2 such that if n = (n(1), n(2)) n(1) ≥ c 1 N log N and n(2) ≥ c 2 N · B log N Given n(1) independent samples y n(1) from distribution p(1) and n(2) independent samples y n(2) from distribution p(2), there exists an estimatorf for estimating KL divergence KL(p(1), p(2)) that satisfies, P(|KL(p(1), p(2)) −f (y n(1) , y n(2) )| ≥ ) ≤ exp −2 2 min{n(1), n(2)} 1−2α Theorem E.21 ([Das], [BPA97]). Let d > 1, and n = (n(1), .…”

Section: E6 Optimal Sample Complexity For Kl Divergencementioning

confidence: 99%

Efficient profile maximum likelihood for universal symmetric property estimation

Charikar

Shiragur

Sidford

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

Estimating symmetric properties of a distribution, e.g. support size, coverage, entropy, distance to uniformity, are among the most fundamental problems in algorithmic statistics. While each of these properties have been studied extensively and separate optimal estimators are known for each, in striking recent work, Acharya et al. [ADOS16] showed that there is a single estimator that is competitive for all symmetric properties. This work proved that computing the distribution that approximately maximizes profile likelihood (PML), i.e. the probability of observed frequency of frequencies, and returning the value of the property on this distribution is sample competitive with respect to a broad class of estimators of symmetric properties. Further, they showed that even computing an approximation of the PML suffices to achieve such a universal plug-in estimator. Unfortunately, prior to this work there was no known polynomial time algorithm to compute an approximate PML and it was open to obtain a polynomial time universal plug-in estimator through the use of approximate PML.In this paper we provide a algorithm (in number of samples) that, given n samples from a distribution, computes an approximate PML distribution up to a multiplicative error of exp(n 2/3 poly log(n)) in time nearly linear in n. Generalizing work of [ADOS16] on the utility of approximate PML we show that our algorithm provides a nearly linear time universal plug-in estimator for all symmetric functions up to accuracy = Ω(n −0.166 ). Further, we show how to extend our work to provide efficient polynomial-time algorithms for computing a d-dimensional generalization of PML (for constant d) that allows for universal plug-in estimation of symmetric relationships between distributions.

show abstract

“…al. [24], we know that n −3/4 log p 3 (n) → α 3 , a constant. Extending a rigorously proved result for plane partitions (d = 2) [26], we also anticipate that the fourdimensional Ferrers diagram of a random solid partition, at large n, will extend the typical distance ≡ n 1/4 in all four directions symmetrically.…”

Section: Fitting the Datamentioning

confidence: 99%

“…Even though the generating function for solid partitions is not known, it has been shown that n −(d+1)/d log p d (n) has a finite limit [24], that we denote by α d . It has later been proposed that even though incorrect, MacMahon's conjecture might be asymptotically exact, thus providing the Ansatz value α 3 1.78982 in addition to values for coefficients of the sub-leading terms [8,22].…”

Section: Introductionmentioning

confidence: 99%

Estimating the Asymptotics of Solid Partitions

Destainville

Govindarajan

2014

J Stat Phys

View full text Add to dashboard Cite

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.

show abstract

Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals

Cited by 18 publications

References 3 publications

Numerical estimation of the asymptotic behaviour of solid partitions of an integer

Numerical estimation of the asymptotic behaviour of solid partitions of an integer

Efficient profile maximum likelihood for universal symmetric property estimation

Estimating the Asymptotics of Solid Partitions

Contact Info

Product

Resources

About