scholarly journals Proof of the Bessenrodt–Ono Inequality by Induction

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
Keyword(s):  
Partition Function ◽  
Residue Class ◽  
Recursion Formula ◽  
Circle Method ◽  
Asymptotic Formulas ◽  
Combinatorial Proof ◽  
Colored Partitions ◽  
Regular Partitions

AbstractIn 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern–Fu–Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.

scholarly journals From partitions to Hodge numbers of Hilbert schemes of surfaces

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180435 ◽  
Author(s):  
Nate Gillman ◽  
Xavier Gonzalez ◽  
Ken Ono ◽  
Larry Rolen ◽  
Matthew Schoenbauer
Keyword(s):  
Partition Function ◽  
Royal Society ◽  
Asymptotic Properties ◽  
Circle Method ◽  
100Th Anniversary ◽  
Hilbert Schemes ◽  
Hodge Numbers ◽  
Projective Surfaces

We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, marking the birth of the ‘circle method’, we present a contemporary example of its legacy in topology. We deduce the equidistribution of Hodge numbers for Hilbert schemes of suitable smooth projective surfaces. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

scholarly journals Manin's conjecture for certain biprojective hypersurfaces

2016 ◽  
Vol 2016 (714) ◽  
Author(s):  
Damaris Schindler
Keyword(s):  
Circle Method ◽  
Height Function ◽  
Singular Locus ◽  
Rational Points ◽  
Large Dimension ◽  
Side Length ◽  
Asymptotic Formulas ◽  
Complete Intersections ◽  
Integer Points ◽  
Manin’S Conjecture

AbstractUsing the circle method, we count integer points on complete intersections in biprojective space in boxes of different side length, provided the number of variables is large enough depending on the degree of the defining equations and certain loci related to the singular locus. Having established these asymptotics we deduce asymptotic formulas for rational points on such varieties with respect to the anticanonical height function. In particular, we establish a conjecture of Manin for certain smooth hypersurfaces in biprojective space of sufficiently large dimension.

scholarly journals Overpartitions with Restricted Odd Differences

10.37236/5248 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jehanne Dousse ◽  
Jeremy Lovejoy ◽  
Karl Mahlburg
Keyword(s):  
Modular Form ◽  
Generating Function ◽  
Difference Equations ◽  
Circle Method ◽  
Hecke Operators ◽  
Asymptotic Formulas ◽  
Linear Congruences ◽  
The Difference ◽  
Mock Modular Form

We use $q$-difference equations to compute a two-variable $q$-hypergeometric generating function for overpartitions where the difference between two successive parts may be odd only if the larger part is overlined. This generating function specializes in one case to a modular form, and in another to a mixed mock modular form. We also establish a two-variable generating function for the same overpartitions with odd smallest part, and again find modular and mixed mock modular specializations. Applications include linear congruences arising from eigenforms for $3$-adic Hecke operators, as well as asymptotic formulas for the enumeration functions. The latter are proven using Wright's variation of the circle method.

A combinatorial proof of a recurrence relation for the sum of divisors function

2020 ◽  
pp. 1-7
Author(s):  
Sun Kim
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Formula ◽  
Combinatorial Proof ◽  
Sum Of Divisors

We give a combinatorial proof of a generalization of an identity involving the sum of divisors function [Formula: see text] and the partition function [Formula: see text] which is a companion of Euler’s recurrence formula for [Formula: see text]

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.)

Asymptotic formulas for coefficients of Kac–Wakimoto Characters

2013 ◽  
Vol 155 (1) ◽  
pp. 51-72 ◽  
Author(s):  
KATHRIN BRINGMANN ◽  
KARL MAHLBURG
Keyword(s):  
Modular Forms ◽  
Asymptotic Series ◽  
Circle Method ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Saddle Point Method ◽  
Asymptotic Formulas ◽  
Affine Lie Algebras ◽  
Maass Forms ◽  
Mock Modular Forms

AbstractWe study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebrassℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

scholarly journals Polynomization of the Bessenrodt–Ono Inequality

2020 ◽  
Vol 24 (4) ◽  
pp. 697-709
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser ◽  
Robert Tröger
Keyword(s):  
Number Theory ◽  
Partition Function ◽  
Real Numbers ◽  
Special Values ◽  
Colored Partitions ◽  
Roots Of Polynomials

Abstract In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials $$P_n(x)$$ P n ( x ) . We prove for all real numbers $$x >2 $$ x > 2 and $$a,b \in \mathbb {N}$$ a , b ∈ N with $$a+b >2$$ a + b > 2 the inequality: $$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$ P a ( x ) · P b ( x ) > P a + b ( x ) . We show that $$P_n(x) < P_{n+1}(x)$$ P n ( x ) < P n + 1 ( x ) for $$x \ge 1$$ x ≥ 1 , which generalizes $$p(n) < p(n+1)$$ p ( n ) < p ( n + 1 ) , where p(n) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: $$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$ P 2 ( - 3 + 10 ) = P 3 ( - 3 + 10 ) .

scholarly journals CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS

2013 ◽  
Vol 09 (08) ◽  
pp. 1885-1894 ◽  
Author(s):  
SU-PING CUI ◽  
NANCY SHAN SHAN GU
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Partition Functions ◽  
Arithmetic Properties ◽  
Partition Analysis ◽  
Congruence Relations ◽  
Modulo P ◽  
Regular Partitions

Andrews and Paule introduced broken k-diamond partitions by using MacMahon's partition analysis. Recently, Fu found a generalization which he called k dots bracelet partitions and investigated some congruences for this kind of partitions. In this paper, by finding congruence relations between the generating function for 5 dots bracelet partitions and that for 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime p, we study arithmetic properties modulo p of k dots bracelet partitions.

THE RESTRICTED 3-COLORED PARTITION FUNCTION MOD 3

2013 ◽  
Vol 09 (07) ◽  
pp. 1789-1799
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Modular Forms ◽  
Type Identity ◽  
Colored Partitions ◽  
Combinatorial Interpretations

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 ( mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.

scholarly journals Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser

2021 ◽  
Vol 8 (21) ◽  
pp. 615-634
Author(s):  
Kathrin Bringmann ◽  
Ben Kane ◽  
Larry Rolen ◽  
Zack Tripp
Keyword(s):  
Partition Function ◽  
Future Study ◽  
Partition Functions ◽  
Colored Partitions ◽  
The Future ◽  
Error Terms ◽  
Partition Inequalities ◽  
User Friendly

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

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