Karl Mahlburg

The overpartition function modulo small powers of 2

Abstract In a recent paper, Fortin et al. (Jagged Partitions, arXivmath. CO/0310079, 2003) found congruences modulo powers of 2 for the values of the overpartition function p ¯ ( n ) in arithmetic progressions. The moduli for these congruences ranged as high as 64. This note shows that p ¯ ( n ) ≡ 0 ( mod 64 ) for a set of integers of arithmetic density 1.

Partition congruences and the Andrews-Garvan-Dyson crank

In 1944, Freeman Dyson conjectured the existence of a “crank” function for partitions that would provide a combinatorial proof of Ramanujans congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

Asymptotic inequalities for positive crank and rank moments

Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases. Our main result states that the two families of moment functions are asymptotically equal, but the crank moments are also asymptotically larger than the rank moments. Andrews, Chan, and Kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt-function. Our main results therefore also give the asymptotic behavior of the ospt-function (and its analogs for higher moments), and we further determine the behavior of the ospt-function modulo 2 by relating its parity to Andrews’ spt-function.

Asymptotics for rank and crank moments

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture of two of the authors that refined a conjecture of Garvan. Garvans original conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The refined version provides precise asymptotic estimates for both the moments and their differences. Our proof uses the Hardy–Ramanujan Circle Method, multiple sums of Bernoulli polynomials and the theory of quasimock theta functions.

Taylor coefficients of mock-Jacobi forms and moments of partition statistics

We develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms. As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

High density piecewise syndeticity of sumsets

Abstract Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A + B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B , there is a high density set of witnesses to the piecewise syndeticity of A + B . Most of the results are shown to hold more generally for subsets of Z d . The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.

Double series representations for Schur's partition function and related identities

We prove new double summation hypergeometric q-series representations for several families of partitions, including those that appear in the famous product identities of Gollnitz, Gordon, and Schur. We give several different proofs for our results, using bijective partitions mappings and modular diagrams, the theory of q-difference equations and recurrences, and the theories of summation and transformation for q-series. We also consider a general family of similar double series and highlight a number of other interesting special cases.

Asymptotic formulas for coefficients of Kac–Wakimoto Characters

We study the coefficients of Kac and Wakimoto’s character formulas for the affine Lie superalgebras s (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.

Partition identities and a theorem of Zagier

In the study of partition theory and q-series, identities that relate series to infinite products are of great interest (such as the famous Rogers-Ramanujan identities). Using a recent result of Zagier, we obtain an infinite family of such identities that is indexed by the positive integers. For example, if m = 1, then we obtain the classical Eisenstein series identity Σλ≥1 odd (-1)(λ-1)/2qλ/(1-q2λ)=q Πn=1∞ (1-q8n)4/(1-q4n)2 If m = 2 and (·/3;) denotes the usual Legendre symbol modulo 3, then we obtain Σλ ≥1 (λ/3) qλ/(1-q2λ=q Πn=1∞ (1-qn)(1-q6n)6/(1-q2n)2(1-q3n)3 We describe some of the partition theoretic consequences of these identities. In particular, we find simple formulas that solve the well-known problem of counting the number of representations of an integer as a sum of an arbitrary number of triangular numbers.

Random Approaches to Fibonacci Identities

Many combinatorialists live by Mach’s words, and take it as a personal challenge. For example, nearly all of the Fibonacci identities in [5] and [6] have been explained by counting arguments [1, 2, 3]. Among the holdouts are those involving infinite sums and irrational quantities. However, by adopting a probabilistic viewpoint, many of the remaining identities can be explained combinatorially. As we shall demonstrate, even the “irrational-looking” Binet’s formula for the n-th Fibonacci number