Frank G. Garvan

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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) in the same way that (1.1) and (1.2) treat the primes 5 and 7. In [4, 5], one of us was able to find a crank relative to vector partitions as follows: For a(More)
Andrews’ 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 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)
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 congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain an explicit(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 Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. In 2008, the first author found Ramanujantype congruences for the spt-function mod 5, 7(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 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: p0(5n+ 4) ≡ p2(5n+ 4) ≡ 0 (mod(More)
In 2003, Hammond and Lewis defined a statistic on partitions into 2 colors which combinatorially explains certain well known partition congruences mod 5. We give two analogs of Hammond and Lewis’s birank statistic. One analog is in terms of Dyson’s rank and the second uses the 5-core crank due to Garvan, Kim and Stanton. We discuss Andrews’s bicrank(More)