holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6)â€¦ (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 theâ€¦ (More)

Andrewsâ€™ spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given forâ€¦ (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 surprisingâ€¦ (More)

We utilize Dysonâ€™s concept of the adjoint of a partition to derive new polynomial analogues of Eulerâ€™s Pentagonal Number Theorem. We streamline Dysonâ€™s bijection relating partitions with crank â‰¤ kâ€¦ (More)

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic srank(Ï€) = O(Ï€) âˆ’O(Ï€â€²), where O(Ï€) denotes the number of odd parts of the partition Ï€ and Ï€â€²â€¦ (More)

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 Ramanujanâ€™s partitionâ€¦ (More)

New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujanâ€™s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for theâ€¦ (More)

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dysonâ€™s rank statistic. He used these connections to find identitiesâ€¦ (More)

Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruentâ€¦ (More)