Dyson's Rank Research Articles

Eventual log-concavity of k-rank statistics for integer partitions

Let Nk(m,n) denote the number of partitions of n with Garvan k-rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k-rank for k=1 and k=2, respectively. In this paper, we prove that the sequences (Nk(m,n))|m|≤n−k−71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently.

Proofs of two conjectural Andrews–Beck type congruences due to Lin, Peng and Toh

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of [Formula: see text] with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for [Formula: see text]-colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo [Formula: see text] by utilizing some [Formula: see text]-series techniques and the theory of modular forms.

A new approach to the Dyson rank conjectures

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of $$5n+4$$ into five classes of equal size. This gave a combinatorial explanation of Ramanujan’s famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of $$7n+5$$ . In 1954 Atkin and Swinnerton-Dyer proved Dyson’s rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this paper we describe a new and more elementary approach using Hecke–Rogers series.

Metrological Detection of Multipartite Entanglement from Young Diagrams.

We characterize metrologically useful multipartite entanglement by representing partitions with Young diagrams. We derive entanglement witnesses that are sensitive to the shape of Young diagrams and show that Dyson's rank acts as a resource for quantum metrology. Common quantifiers, such as the entanglement depth and k-separability are contained in this approach as the diagram's width and height. Our methods are experimentally accessible in a wide range of atomic systems, as we illustrate by analyzing published data on the quantum Fisher information and spin-squeezing coefficients.

Asymptotic equidistribution and convexity for partition ranks

We study the Dyson rank function $N(r,t;n)$, the number of partitions with rank congruent to $r$ modulo $t$. We first show that it is monotonic in $n$, and then show that it equidistributed as $n \rightarrow \infty$. Using this result we prove a conjecture of Hou and Jagadeeson on the convexity of $N(r,t;n)$.

Mock modularity of the Md-rank of overpartitions

We investigate the modular properties of a new partition rank, the Md-rank of overpartitions. In fact this is an infinite family of ranks, indexed by the positive integer d, that gives both the Dyson rank of overpartitions and the overpartition M2-rank as special cases. The Md-rank of overpartitions is the holomorphic part of a certain harmonic Maass form of weight 12. We give the exact transformation of this harmonic Maass form along with a few identities for the Md-rank.

Transformation properties for Dyson’s rank function

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(\zeta,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.

Dyson’s partition ranks and their multiplicative extensions

We study the Dyson rank function $N(r,3;n)$, the number of partitions of $n$ with rank $\equiv r \pmod 3$. We investigate the convexity of these functions. We extend $N(r,3;n)$ multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size $n$.

On three theorems of Folsom, Ono and Rhoades

In his deathbed letter to Hardy, Ramanujan gave a vague definition of a mock modular function: at each root of unity its asymptotics matches the one of a modular form, though a choice of the modular function depends on the root of unity. Recently Folsom, Ono and Rhoades have proved an elegant result about the match for a general family related to Dyson's rank (mock theta) function and the Andrews--Garvan crank (modular) function---the match with explicit formulae for implied $O(1)$ constants. In this note we give another elementary proof of Ramanujan's original claim and outline some heuristics which may be useful for obtaining a new proof of the general Folsom--Ono--Rhoades theorem.

Universal mock theta functions and two-variable Hecke–Rogers identities

We obtain two-variable Hecke–Rogers identities for three universal mock theta functions. This implies that many of Ramanujan’s mock theta functions, including all the third-order functions, have a Hecke–Rogers-type double sum representation. We find new generating function identities for the Dyson rank function, the overpartition rank function, the \(M2\)-rank function and related spt-crank functions. Results are proved using the theory of basic hypergeometric functions.

Hecke operators on certain subspaces of integral weight modular forms

Recent works of F. G. Garvan ([Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank, Int. J. Number Theory6(12) (2010) 281–309; MR2646759 (2011j:05032)]) and Y. Yang ([Congruences of the partition function, Int. Math. Res. Not.2011(14) (2011) 3261–3288; MR2817679 (2012e:11177)] and [Modular forms for half-integral weights on SL 2(ℤ), to appear in Nagoya Math. J.]) concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL 2(ℤ)-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL 2(ℤ)-forms of fixed weight.

The spt-crank for overpartitions

Bringmann, Lovejoy, and Osburn (14, 15) showed that the generating functions of the spt- overpartition functions spt (n), spt1 (n), spt2 (n), and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (7) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences mod 5 and 7 for spt(n). Chen, Ji, and Zang (17) were able to define this spt-crank in terms of ordinary partitions. In this paper we define spt-cranks in terms of vector partitions that combinatorially explain the known simple congruences for all the spt-overpartition functions as well as new simple congruences. For all the overpartition functions except M2spt(n) we are able to define the spt-crank purely in terms of marked overpartitions. The proofs of the congruences depend on Bailey's Lemma and the difference formulas for the Dyson rank of an overpartition (24) and the M2-rank of a partition without repeated odd parts (25).

Combinatorial interpretations of congruences for the spt-function

Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. In 1988, the second author gave new combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. In 2008, the first author found Ramanujan-type congruences for the spt-function mod 5, 7 and 13. We give new combinatorial interpretations of the spt-congruences mod 5 and 7. These are in terms of the same crank but for a restricted set of vector partitions. The proof depends on relating the spt-crank with the crank of vector partitions and the Dyson rank of ordinary partitions. We derive a number of identities for spt-crank modulo 5 and 7. We prove the surprising result that all the spt-crank coefficients are nonnegative.

The combinatorics of $k$-marked Durfee symbols

Andrews recently introduced k-marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a relation between k-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on k-marked Durfee symbols given by Andrews could follow from this relation. In a similar manner, we also prove the identities due to Andrews on k-marked odd Durfee symbols combinatorially, which resemble ordinary k-marked Durfee symbols with a modified subscript and with odd numbers as entries.

Dyson's Rank, Overpartitions, and Weak Maass Forms

In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decompositions of Ramanujan-type congruences for overpartitions as well as the modularity of rank differences in certain arithmetic progressions.

CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

Number theoretic properties of generating functions related to Dyson’s rank for partitions into distinct parts

Let Q(n) denote the number of partitions of n into distinct parts. We show that Dyson's rank provides a combinatorial interpretation of the well-known fact that Q(n) is almost always divisible by 4. This interpretation gives rise to a new false theta function identity that reveals surprising analytic properties of one of Ramanujan's mock theta functions, which in turn gives generating functions for values of certain Dirichlet L-functions at nonpositive integers.

CONGRUENCES FOR DYSON'S RANKS

In this paper, we prove infinite families of congruences for coefficients of harmonic Maass forms whose coefficients encode Dyson's rank. This generalizes the earlier joint work of the author with Ken Ono.

Partition Statistics and Quasiharmonic Maass Forms

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating to their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between them. In this paper, we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of harmonic Maass forms and their derivatives. In particular, these generating functions may be viewed as analogs of Ramanujan's mock theta functions with arbitrarily high weight. We use the automorphic properties to prove the existence of infinitely many congruences for the Durfee symbols. Furthermore, we show that as k varies, the modularity of the k-marked Durfee symbols is precisely dictated by the case k = 2. Finally, we use this relation in order to prove the existence of general congruences for rank moments in terms of level one modular forms of bounded weight.

RANK AND CONGRUENCES FOR OVERPARTITION PAIRS

The rank of an overpartition pair is a generalization of Dyson's rank of a partition. The purpose of this paper is to investigate the role that this statistic plays in the congruence properties of [Formula: see text], the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented.