Partition Statistics and Quasiharmonic Maass Forms

Andrews recently introduced k-marked Durfee symbols, which area generalization of partitions that are connected to momentsof Dyson's rank statistic. He used these connections to findidentities relating to their generating functions as well asto prove Ramanujan-type congruences for these objects and findrelations between them. In this paper, we show that the hypergeometricgenerating functions for these objects are natural examplesof quasimock theta functions, which are defined as the holomorphicparts of harmonic Maass forms and their derivatives. In particular,these generating functions may be viewed as analogs of Ramanujan'smock theta functions with arbitrarily high weight. We use theautomorphic properties to prove the existence of infinitelymany congruences for the Durfee symbols. Furthermore, we showthat as k varies, the modularity of the k-marked Durfee symbolsis precisely dictated by the case k = 2. Finally, we use thisrelation in order to prove the existence of general congruencesfor rank moments in terms of level one modular forms of boundedweight.     

Higher order spt-functions

Andrews? spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only known previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.     

Detection and imaging of He2 molecules in superfluid helium

We present data that show a cycling transition can be used to detect and image metastable He2 triplet molecules in superfluid helium. We demonstrate that limitations on the cycling efficiency due to the vibrational structure of the molecule can be mitigated by the use of repumping lasers. Images of the molecules obtained using the method are also shown. This technique gives rise to a new kind of ionizing radiation detector. The use of He2 triplet molecules as tracer particles in the superfluid promises to be a powerful tool for visualization of both quantum and classical turbulence in liquid helium.     

Relations Between the Ranks and Cranks of Partitions

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.     

Dissecting the Stanley partition function

Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p_i(n) denote the number of partitions π of n such that . Here denotes the number of odd parts of the partition π and π^′ is the conjugate of π. Stanley [Amer. Math. Monthly 109 (2002) 760; Adv. Appl. Math., to appear] derived an infinite product representation for the generating function of p₀(n)-p₂(n). Recently, Swisher [The Andrews–Stanley partition function and p(n), preprint, submitted for publication] employed the circle method to show that

(i)

and that for sufficiently large n

(ii)

In this paper we study the even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result:

Two proofs of this surprising inequality are given. The first one uses the Göllnitz–Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates.     

Combinatorial interpretations of Ramanujan’s tau function

We use a $q$-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves $t$-cores and a new class of partitions which we call $(m,k)$-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.