scholarly journals Partitions and the Maximal Excludant

10.37236/8736 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Shane Chern
Keyword(s):  
Generating Function ◽  
Theta Function ◽  
Nonnegative Integer ◽  
Mock Theta Function ◽  
Integer Partition ◽  
The Difference

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ as the largest nonnegative integer smaller than the largest part of $\pi$ that is not itself a part. Let $\sigma\!\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\sigma\!\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts over all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\to \infty$.

scholarly journals Congruences modulo powers of 5 for the rank parity function

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Dandan Chen ◽  
Rong Chen ◽  
Frank Garvan
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Theta Function ◽  
Mock Theta Function ◽  
Parity Function ◽  
International Audience

International audience It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for the rank parity function is f (q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.

A mock theta function identity related to the partition rank modulo 3 and 9

2020 ◽  
pp. 1-17
Author(s):  
Song Heng Chan ◽  
Nankun Hong ◽  
Jerry ◽  
Jeremy Lovejoy
Keyword(s):  
Generating Function ◽  
Theta Function ◽  
Generating Functions ◽  
Mock Theta Function ◽  
Function Identity ◽  
Theta Function Identity ◽  
Cranks Of Partitions

We prove a new mock theta function identity related to the partition rank modulo 3 and 9. As a consequence, we obtain the [Formula: see text]-dissection of the rank generating function modulo [Formula: see text]. We also evaluate all of the components of the rank–crank differences modulo [Formula: see text]. These are analogous to conjectures of Lewis [The generating functions of the rank and crank modulo 8, Ramanujan J. 18 (2009) 121–146] on rank–crank differences modulo [Formula: see text], first proved by Mortenson [On ranks and cranks of partitions modulo 4 and 8, J. Combin. Theory Ser. A 161 (2019) 51–80].

OVERPARTITIONS RELATED TO THE MOCK THETA FUNCTION

2020 ◽  
Vol 102 (3) ◽  
pp. 410-417
Author(s):  
BERNARD L. S. LIN
Keyword(s):  
Generating Function ◽  
Theta Function ◽  
Short Proof ◽  
Mock Theta Function ◽  
Eighth Order ◽  
Order Mock Theta Function

Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number $v_{0}(n)$ of overpartitions of $n$ into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function $V_{0}(q)$ of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of $v_{0}(2n)$. Then we establish three congruences for $v_{0}(n)$ along certain progressions which are subsequences of the integers $4n+3$.

scholarly journals Higher depth mock theta functions and q-hypergeometric series

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joshua Males ◽  
Andreas Mono ◽  
Larry Rolen
Keyword(s):  
Theta Function ◽  
Modular Forms ◽  
Hypergeometric Series ◽  
Theta Functions ◽  
Mock Theta Function ◽  
Mock Theta Functions ◽  
Maass Forms ◽  
Mock Modular Forms ◽  
Harmonic Maass Forms

Abstract In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals A Mock Theta Function of Second Order

2009 ◽  
Vol 2009 ◽  
pp. 1-15
Author(s):  
Bhaskar Srivastava
Keyword(s):  
Unit Circle ◽  
Theta Function ◽  
Theta Series ◽  
Mathematical Physics ◽  
Second Order ◽  
Mock Theta Function ◽  
Third Order ◽  
Quantum Invariant ◽  
Order Mock Theta Function

We consider the second-order mock theta function (), which Hikami came across in his work on mathematical physics and quantum invariant of three manifold. We give their bilateral form, and show that it is the same as bilateral third-order mock theta function of Ramanujan. We also show that the mock theta function () outside the unit circle is a theta function and also write as a coefficient of of a theta series. First writing as a coefficient of a theta function, we prove an identity for .

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.

Sign in / Sign up

Export Citation Format

Share Document