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1. Introduction. In [3], F. J. Dyson defined the rank of a partition as the largest part minus the number of parts. He let 7V(ra, t, n) denote the number of partitions of n of rank congruent to m modulo £, and he conjectured

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 their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between. In this paper we show that the… (More)

- F. G. GARVAN
- 2010

Andrews' spt-function can be written as the difference between the second sym-metrized 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… (More)

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit Ramanujan-type congruences for spt(n) modulo for all primes ≥ 5 which were conjectured earlier by the author. We extend… (More)

- BRUCE C. BERNDT, FRANK G. GARVAN, F. G. GARVAN, ü-.fiß
- 1995

In his famous paper on modular equations and approximations to n , Ramanujan offers several series representations for X/n, which he claims are derived from "corresponding theories" in which the classical base q is replaced by one of three other bases. The formulas for \fn were only recently proved by J. M. and P. B. Borwein in 1987, but these… (More)

- F. G. GARVAN
- 2008

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 con-gruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain an… (More)

- Alexander Berkovich, Frank G. Garvan
- J. Comb. Theory, Ser. A
- 2002

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank ≤ 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… (More)

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 their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between. In this paper we show that the… (More)

- BEAUTIFUL IDENTITY, HEI-CHI CHAN, +6 authors Frank Garvan
- 2008

In this paper, we prove a generalization of Ramanu-jan's " Most Beautiful Identity. " Our generalization is closely related to Ramanujan's beautiful results involving the cubic continued fraction.

Many of Ramanujan's ideas and theorems form the seeds of questions and problems, many of which remain unresolved or even to be thoroughly examined. This survey raises questions arising from Ramanujan's work on theta-functions and other q-series, with Gaussian hypergeometric functions making frequent appearances.