Asymptotic Formulaæ in Combinatory Analysis

  title={Asymptotic Formula{\ae} in Combinatory Analysis},
  author={G. Hardy and S. Ramanujan},
  journal={Proceedings of The London Mathematical Society},
m -ARY Partitions
We study the number of ways in which n ≥ 0 be written as the sum of powers of m ≥ 2. After briefly discussing historical results and examples, we prove recurrence relations, exact formulae, bounds,Expand
Will the real Hardy-Ramanujan formula please stand up?
Five score and several years ago, G. H. Hardy and S. Ramanujan wrote a consequential paper titled Asymptotic Formulae in Combinatory Analysis.
Partitions: At the Interface of q-Series and Modular Forms
In this paper we explore five topics from the theory of partitions: (1) the Rademacher conjecture, (2) the Herschel-Cayley-Sylvester formulas, (3) the asymptotic expansions of E.M. Wright, (4) theExpand
Differences of the partition function
Let p(n) denote the number of unrestricted partitions of n, and let ∆p(n) = p(n) − p(n − 1 ), ∆k p(n) = ∆(∆k − 1 p) (n). This note answers several questions about the behavior of the k-difference ∆kExpand
Homomorphisms from the torus
We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that theExpand
Maximum entropy and integer partitions
We derive asymptotic formulas for the number of integer partitions with given sums of jth powers of the parts for j belonging to a finite, non-empty set J ⊂ N. The method we use is based on theExpand
Profile Entropy: A Fundamental Measure for the Learnability and Compressibility of Discrete Distributions
The profile of a sample is the multiset of its symbol frequencies. We show that for samples of discrete distributions, profile entropy is a fundamental measure unifying the concepts of estimation,Expand
Profile Entropy: A Fundamental Measure for the Learnability and Compressibility of Discrete Distributions
To further the understanding of profile entropy, its attributes are investigated, algorithms for approximating its value are provided, and its magnitude for numerous structural distribution families is determined. Expand
Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields
Heuristics of Cohen–Lenstra together with the prediction for the asymptotic behavior of Soundararajan’s conjecture are combined to make precise predictions about the ascyptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups. Expand
Lifshitz scaling, microstate counting from number theory and black hole entropy
A bstractNon-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation E ∼ kz and dynamical exponent z > 1. The asymptotic growthExpand