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 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.

Reversible coagulation-fragmentation processes and random combinatorial structures: Asymptotics for the number of groups

2004 ◽  
Vol 25 (2) ◽  
pp. 227-245 ◽  
Author(s):  
Michael M. Erlihson ◽  
Boris L. Granovsky
Keyword(s):  
Combinatorial Structures ◽  
Reversible Coagulation ◽  
Fragmentation Processes

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.)

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

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 Effective bounds for the Andrews spt-function

2019 ◽  
Vol 31 (3) ◽  
pp. 743-767 ◽  
Author(s):  
Madeline Locus Dawsey ◽  
Riad Masri
Keyword(s):  
Partition Function ◽  
Asymptotic Formula ◽  
Error Bounds ◽  
Error Term ◽  
Conditional Convergence ◽  
Type Formula ◽  
Singular Moduli ◽  
Smallest Parts Function

Abstract In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function {{\mathrm{spt}}(n)} . We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function {p(n)} and {{\mathrm{spt}}(n)} . Further, we strengthen one of the conjectures, and prove that for every {\epsilon>0} there is an effectively computable constant {N(\epsilon)>0} such that for all {n\geq N(\epsilon)} , we have \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<{\mathrm{spt}}(n)<\bigg{(}\frac{\sqrt{6}}{% \pi}+\epsilon\bigg{)}\sqrt{n}\,p(n). Due to the conditional convergence of the Rademacher-type formula for {{\mathrm{spt}}(n)} , we must employ methods which are completely different from those used by Lehmer to give effective error bounds for {p(n)} . Instead, our approach relies on the fact that {p(n)} and {{\mathrm{spt}}(n)} can be expressed as traces of singular moduli.

PARTITIONS WITH PARTS IN A FINITE SET

2006 ◽  
Vol 02 (03) ◽  
pp. 455-468 ◽  
Author(s):  
ØYSTEIN J. RØDSETH ◽  
JAMES A. SELLERS
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Residue Class ◽  
Finite Set ◽  
Positive Integers

For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.

scholarly journals An asymptotic formula for the t-core partition function and a conjecture of Stanton

2008 ◽  
Vol 128 (9) ◽  
pp. 2591-2615 ◽  
Author(s):  
Jaclyn Anderson
Keyword(s):  
Partition Function ◽  
Asymptotic Formula

Congruence properties of the binary partition function

1969 ◽  
Vol 66 (2) ◽  
pp. 371-376 ◽  
Author(s):  
R. F. Churchhouse
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Functional Equations ◽  
Precise Calculation ◽  
Binary Partition ◽  
Powers Of 2 ◽  
De Bruijn ◽  
Congruence Properties

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

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