Title

Combinatorial interpretations of congruences for the spt-function

Authors

Authors

G. E. Andrews; F. G. Garvan;J. Liang

Comments

Authors: contact us about adding a copy of your work at STARS@ucf.edu

Abbreviated Journal Title

Ramanujan J.

Keywords

Spt-function; Partitions; Rank; Crank; Vector partitions; Ramanujan's; Lost Notebook; Congruences; Basic hypergeometric series; PARTITIONS; Mathematics

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.

Journal Title

Ramanujan Journal

Volume

29

Issue/Number

1-3

Publication Date

1-1-2012

Document Type

Article

Language

English

First Page

321

Last Page

338

WOS Identifier

WOS:000310975300021

ISSN

1382-4090

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