A Note on the Exact Expected Length of the kth Part of a Random Partition

Integers ◽  
2010 ◽  
Vol 10 (3) ◽  
Author(s):  
Kimmo Eriksson
Keyword(s):  
Partition Function ◽  
Asymptotic Formula ◽  
Exact Formula ◽  
Weighted Sum ◽  
Random Partition ◽  
Bijective Proof ◽  
Expected Length

AbstractKessler and Livingstone proved an asymptotic formula for the expected length of the largest part of a partition drawn uniformly at random. As a first step they gave an exact formula expressed as a weighted sum of Euler's partition function. Here we give a short bijective proof of a generalization of this exact formula to the expected length of the

On the kth root partition function

2021 ◽  
pp. 1-15
Author(s):  
Ya-Li Li ◽  
Jie Wu
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Real Number ◽  
Number Of Solutions ◽  
Asymptotic Development ◽  
Number Formula

For any positive integer [Formula: see text], let [Formula: see text] be the number of solutions of the equation [Formula: see text] with integers [Formula: see text], where [Formula: see text] is the integral part of real number [Formula: see text]. Recently, Luca and Ralaivaosaona gave an asymptotic formula for [Formula: see text]. In this paper, we give an asymptotic development of [Formula: see text] for all [Formula: see text]. Moreover, we prove that the number of such partitions is even (respectively, odd) infinitely often.

A Risk Model with Delayed Claims

2013 ◽  
Vol 50 (03) ◽  
pp. 686-702 ◽  
Author(s):  
Angelos Dassios ◽  
Hongbiao Zhao
Keyword(s):  
Asymptotic Formula ◽  
Poisson Process ◽  
Ruin Probability ◽  
Risk Model ◽  
Poisson Model ◽  
Exact Formula ◽  
Poisson Models ◽  
Asymptotic Expressions ◽  
Special Case

In this paper we introduce a simple risk model with delayed claims, an extension of the classical Poisson model. The claims are assumed to arrive according to a Poisson process and claims follow a light-tailed distribution, and each loss payment of the claims will be settled with a random period of delay. We obtain asymptotic expressions for the ruin probability by exploiting a connection to Poisson models that are not time homogeneous. A finer asymptotic formula is obtained for the special case of exponentially delayed claims and an exact formula is obtained when the claims are also exponentially distributed.

scholarly journals Power partitions and a generalized eta transformation property

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Don Zagier
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
General Type ◽  
Circle Method ◽  
Exact Formula ◽  
Transformation Property ◽  
Roots Of Unity ◽  
Root Of Unity ◽  
Famous Paper ◽  
The One

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

scholarly journals AN ASYMPTOTIC FORMULA FOR THE COEFFICIENTS OF j(z)

2013 ◽  
Vol 09 (03) ◽  
pp. 641-652 ◽  
Author(s):  
MICHAEL DEWAR ◽  
M. RAM MURTY
Keyword(s):  
Partition Function ◽  
Asymptotic Formula ◽  
Elliptic Curves ◽  
Circle Method ◽  
Steepest Descent ◽  
Laplace’S Method ◽  
Laplace's Method ◽  
Method Of Steepest Descent

We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant of elliptic curves. Our proof does not use the circle method. We use Laplace's method of steepest descent and the Hardy–Ramanujan asymptotic formula for the partition function. (The latter asymptotic formula can be derived without the circle method.)

On Riesz means of the coefficients of the Rankin–Selberg series

1999 ◽  
Vol 127 (1) ◽  
pp. 117-131 ◽  
Author(s):  
ALEKSANDAR IVIĆ ◽  
KOHJI MATSUMOTO ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Upper Bound ◽  
Error Term ◽  
Weighted Sum ◽  
Mean Square ◽  
Riesz Means ◽  
The Mean ◽  
Voronoi Formula

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

scholarly journals Rademacher expansions and the spectrum of 2d CFT

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Luis F. Alday ◽  
Jin-Beom Bae
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Analytic Number Theory ◽  
Exact Formula ◽  
Conformal Field Theories ◽  
Light Spectrum ◽  
Chiral Algebra ◽  
Positive Weight ◽  
Conformal Field ◽  
Pure Gravity

Abstract A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin j ≠ 0. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

scholarly journals An Asymptotic Formula for Cayley's Double Partition Function $p(2;n)$

1979 ◽  
Vol 02 (1) ◽  
pp. 137-158 ◽  
Author(s):  
Ryuji KANEIWA
Keyword(s):  
Partition Function ◽  
Asymptotic Formula

scholarly journals Asymptotic formula for a partition function of reversible coagulation-fragmentation processes

2002 ◽  
Vol 130 (1) ◽  
pp. 259-279 ◽  
Author(s):  
Gregory A. Freiman ◽  
Boris L. Granovsky
Keyword(s):  
Partition Function ◽  
Asymptotic Formula ◽  
Reversible Coagulation ◽  
Fragmentation Processes

On the Difference of Expected Lengths of Minimum Spanning Trees

2009 ◽  
Vol 18 (3) ◽  
pp. 423-434 ◽  
Author(s):  
WENBO V. LI ◽  
XINYI ZHANG
Keyword(s):  
Spanning Trees ◽  
Minimum Spanning Tree ◽  
Exact Formula ◽  
Expected Length ◽  
Wheel Graphs ◽  
Edge Distribution ◽  
Tutte Polynomials ◽  
Uniform Case ◽  
The Difference ◽  
Identical Edge

An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele's formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, with rate one, and uniform distribution on the interval (0, 1) is shown to be positive and of rate ζ(3)/n. For wheel graphs, precise values of expected lengths are given via calculations of the associated Tutte polynomials.

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