Frank Garvan - Academia.edu (original) (raw)

Papers by Frank Garvan

Advances in Applied Mathematics, Jan 1, 2008

Journal of Combinatorial Theory, Series A, Jan 1, 2002

Constructive Approximation, 1993

ABSTRACT We continue to investigate spt-type functions that arise from Bailey pairs. We prove sim... more ABSTRACT We continue to investigate spt-type functions that arise from Bailey pairs. We prove simple Ramanujan type congruences for these functions which can be explained by a spt-crank-type function. The spt-crank-type functions are constructed by adding an extra variable zzz into the generating functions. We find these generating functions to have interesting representations as either infinite products or as Hecke-Rogers-type double series. These series reduce nicely when zzz is a certain root of unity and allow us to deduce congruences. Additionally we find dissections when zzz is a certain root of unity to explain the congruences. Our double sum and product formulas require Bailey's Lemma and conjugate Bailey pairs. Our dissection formulas follow from Bailey's Lemma and dissections of known ranks and cranks.

Contemporary Mathematics, 1994

International Journal of Number Theory, 2014

ABSTRACT Inspired by recent congruences by N. Andersen [Int. J. Number Theory 9, No. 3, 713–728 (... more ABSTRACT Inspired by recent congruences by N. Andersen [Int. J. Number Theory 9, No. 3, 713–728 (2013; Zbl 1286.11162)] with varying powers of 2 in the modulus for partition related functions, we extend the modulo 32760 congruences of the first author [Congruences for Andrews’ spt -function modulo 32760 and extension of Atkin’s Hecke-type partition congruences. Borwein, Jonathan M. (ed.) et al., Number theory and related fields. In memory of Alf van der Poorten. Springer Proc. Math. Stat. 43, 165–185 (2013; Zbl 1286.11164), Trans. Am. Math. Soc. 364, No. 9, 4847–4873 (2012; Zbl 1286.11165)] for the function spt (n). We show that a normalized form of the generating function of spt (n) is an eigenform modulo 32 for the Hecke operators T(ℓ 2 ) for primes ℓ≥5 with ℓ≡1,11,17,19(mod24), and an eigenform modulo 16 for ℓ≡13,23(mod24).

Transactions of the American Mathematical Society, 1994

The Ramanujan Journal, 2012

ABSTRACT Let spt(n) denote the total number of appearances of the smallest parts in all the parti... more ABSTRACT 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 Ramanujan-type congruences for the spt-function mod 5, 7 and 13. We give new combinatorial interpretations of the spt-congruences mod 5 and 7. These are in terms of the same crank but for a restricted set of vector partitions. The proof depends on relating the spt-crank with the crank of vector partitions and the Dyson rank of ordinary partitions. We derive a number of identities for spt-crank modulo 5 and 7. We prove the surprising result that all the spt-crank coefficients are nonnegative.

The Ramanujan Journal, 2013

Mathematics of Computation, 1990

Journal of Symbolic Computation, 1995