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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)
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)
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)
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)
In this paper, properties of the functions A d (q), B d (q) and C d (q) are derived. Specializing at d = 1 and 2, we construct two new quadratic iterations to π. These are analogues of previous iterations discovered by the Bor-weins (1987), (2002). Two new transformations of the hypergeometric series 2 F 1 (1/3, 1/6; 1; z) are also derived.