Title

Combinatorial Interpretations Of Congruences For The Spt-Function

Keywords

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

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. © 2012 Springer Science+Business Media, LLC.

Publication Date

12-1-2012

Publication Title

Ramanujan Journal

Volume

29

Issue

1-3

Number of Pages

321-338

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s11139-012-9369-7

Socpus ID

84869110375 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84869110375

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