Combinatorial interpretations of congruences for the spt-function

    Authors

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

    Comments

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    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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