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 a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic srank(π) = O(π) − O(π), where O(π) denotes the number of odd parts of the partition π and π is the conjugate of π. In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p 0 (5n + 4) ≡ p 2 (5n + 4) ≡… (More)
We provide elementary proofs of the Farkas–Kra septagonal numbers identity and some kth-order generalizations.
Recently Andrews and Sellers proved some amazing congruences for the Fishburn numbers. We extend their results to a more general sequence of numbers. As a result we prove a new congruence mod 23 for the Fishburn numbers and prove their conjectured mod 5 congruence for a related sequence. We also extend and prove some unpublished conjectures of Garthwaite… (More)
Dedicated to the memory of A.J. (Alf) van der Poorten, my former teacher Abstract. New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part α(24z) of a certain weak Maass form M(z) of weight 3 2. We show that a normalized form of the generating function for spt(n)… (More)
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 O(π)−O(π ′) ≡ i (mod 4). Here O(π) denotes the number of odd parts of the partition π and π ′ is the conjugate of π. R. Stanley ,  derived an infinite product representation for the generating function of p 0 (n) − p… (More)