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In this paper, we build on the recent results of Frank Garvan and Rishabh Sarma as well as classical results of Bruce Berndt in order to establish the 11-dissection of the deviations of the rank and crank modulo 11. Using our new dissections, we re-derive the results of Garvan, Atkin, Swinnerton-Dyer, Hussain, Ekin and Chern. By developing and exploiting positivity conditions for quotients of theta functions, we will also prove new rank–crank inequalities and make several conjectures, one of which was recently solved by Kathrin Bringmann and Badri Vishal Pandey. For other applications of our methods, in this paper, we will also prove new congruences for rank moments as well as the Andrews' smallest parts function and Eisenstein series. [ABSTRACT FROM AUTHOR]

In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n). [ABSTRACT FROM AUTHOR]

In this paper we give a full description of the inequalities that can occur between overpartition ranks modulo c ≥ 2. If N ¯ (a , c , n) denotes the number of overpartitions of n with rank congruent to a modulo c, we prove that for any c ≥ 7 and 0 ≤ a < b ≤ c 2 we have N ¯ (a , c , n) > N ¯ (b , c , n) for n large enough. That the sign of the rank differences N ¯ (a , c , n) - N ¯ (b , c , n) depends on the residue class of n modulo c in the case of small moduli, such as c = 6 , is known due to the work of Ji et al. (J Number Theory 184:235–269, 2018) and Ciolan (Int J Number Theory 16(1):121–143, 2020). We show that the same behavior holds for c ∈ { 2 , 3 , 4 , 5 }. [ABSTRACT FROM AUTHOR]

We introduce a statistic on overpartitions called the k ¯ -rank. When there are no overlined parts, this coincides with the k -rank of a partition introduced by Garvan. Moreover, it reduces to the D-rank of an overpartition when k = 2. The generating function for the k ¯ -rank of overpartitions is given. We also establish a relation between the generating function of self-3-conjugate overpartitions and the tenth-order mock theta functions X(q) and χ (q). [ABSTRACT FROM AUTHOR]

Dyson's famous conjectures (proved by Atkin and Swinnerton-Dyer) gave a combinatorial interpretation of Ramanujan's congruences for the partition function. The proofs of these results center on one of the universal mock theta functions that generate partitions according to Dyson's rank. George Beck has generalized the study of partition function congruences related to rank by considering the total number of parts in the partitions of n. The related generating functions are no longer part of the world of mock theta functions. However, George Beck has conjectured that certain linear combinations of the related enumeration functions do satisfy congruences modulo 5 and 7. The conjectures are proved here. [ABSTRACT FROM AUTHOR]

Lovejoy and Osburn proved formulas for the generating functions for the rank differences of overpartitions modulo 3 and 5. In this paper, we derive formulas for the generating functions for the rank differences of overpartitions modulo 6 and 10. With these generating functions, we obtain some equalities and inequalities on ranks of overpartitions modulo 6 and 10. We also relate these generating functions to the third order mock theta functions ω ( q ) and ρ ( q ) and the tenth order mock theta functions ϕ ( q ) and ψ ( q ) . [ABSTRACT FROM AUTHOR]

In this paper, we prove formulas for the generating functions for the rank and crank differences for partitions modulo 3. In 2000, Andrews and Lewis made conjectures on inequalities satisfied by ranks and cranks modulo 3. These conjectures were first proved by Bringmann and Kane, respectively, using the circle method. Working directly on the generating functions, we obtain improvements to these inequalities. [ABSTRACT FROM AUTHOR]

We present a generalization, which we call ( k, m)-rank, of Dyson’s notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Rogers-Ramanujan identities. We also describe the relationship between ( k, m)-rank and Garvan’s k-rank. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that the Zagier-Eisenstein series shares its nonholomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact formulas, asymptotics, and congruences for the rank differences as well as q-series identities of the mock theta type. [ABSTRACT FROM AUTHOR]

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 $\overline{pp}(n)$, the number of overpartition pairs of n. Some generating functions and identities involving this rank are also presented. [ABSTRACT FROM AUTHOR]

Abstract: We give a combinatorial proof of the first Rogers–Ramanujan identity by using two symmetries of a new generalization of Dyson''s rank. These symmetries are established by direct bijections. [Copyright &y& Elsevier]

Abstract: A review is given of several aspects of the work of Basil Gordon. These include: Rogers–Ramanujan identities, plane partitions, the method of weighted words, modular forms and partition congruences, and the asymptotics of partitions and related q-series. [Copyright &y& Elsevier]

We discuss conjugation and Dyson’s rank for overpartitions from the perspective of the Frobenius representation. More specifically, we translate the classical definition of Dyson’s rank to the Frobenius representation of an overpartition and define a new kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with respect to this conjugation. [ABSTRACT FROM AUTHOR]

We utilize Dyson'' concept of the adjoint of a partition to derive an infinite family of new polynomial analogs of Euler''s Pentagonal Number Theorem. We streamline Dyson''s bijection relating partitions with crank &les;k and those with k in the Rank-Set of partitions. Also, we extend Dyson''s adjoint of a partition to MacMahon''s “modular” partitions with modulus 2. This way we find a new combinatorial proof of Gauss''s famous identity. We give a direct combinatorial proof that for n>1 the partitions of n with crank k are equinumerous with partitions of n with crank −k. [Copyright &y& Elsevier]

In this paper we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions. Using these results, we show that N ‾ (a , c , n) , the number of overpartitions of n with rank congruent to a modulo c , is equidistributed with respect to 0 ≤ a < c , for any c ≥ 2 , as n → ∞ and, in addition, we prove some inequalities between ranks of overpartitions conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018) for n = 6 and n = 10. [ABSTRACT FROM AUTHOR]