Congruence Properties of Andrews’SPT- Function

Authors

  • Sabuj Das Raozan University College

DOI:

https://doi.org/10.18034/abcjar.v3i2.37

Keywords:

Crank, congruences, product notations, Ramanujan -type congruences, spt- function, vector partitions

Abstract

Let spt (n) denote the total number of appearances of the smallest part in each partition of n. In 1988, Garvan gave new combinatorial interpretations of  Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. This paper shows how to generate the generating functions for spt(n),  elaborately and also shows how to prove the relation among the terms spt (n) and. In 2008, Andrews stated Ramanujan- type congruences for the spt- function mod 5, 7 and 13. The new combinatorial  interpretations of the spt- congruences mod 5 and 7 are given in this article. These are in terms of the spt- crank but for a restricted set of vector partitions. The proofs depend on relating the spt- crank with the crank of vector partitions.

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Author Biography

  • Sabuj Das, Raozan University College

    Senior Lecturer, Department of Mathematics, Raozan University College, BANGLADESH

     

References

Andrews GE (1985). The Theory of Partitions, Encyclopedia of Mathematics and its Application, vol. 2 (G.-C. Rota,ed.) Addison-Wesley, Reading, mass, 1976 (Reissued, Cambridge University, Press, London and New York 1985).

Andrews GE, Garvan FG (1988). Dyson’s Crank of a Partition, Bulletin (New series) of the American Mathematical Society, 18(2): 167–171.

Andrews, G. E., The number of smallest parts in the partitions of n, J. Reine Angew. Math . 624, 133-142, (2008).

Baizid AR and Alam MS. 2014. Rciprocal Property of Different Types of Lorentz Transformations International Journal of Reciprocal Symmetry and Theoretical Physics, 1, 20-35.

Garvan FG (1986). Generalization of Dyson’s Rank, Ph.D. Thesis, Pennsylvania State University.

Garvan FG (2013). Dyson’s Rank Function and Andrews’ spt-function, University of Florida, Seminar Paper Presented in the University of Newcastle on 20 August 2013.

Garvan, F.G., New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11, Trans. Am. Math. Soc. 305, 47-77 (1988).

Mohajan HK. 2014. Upper Limit of the Age of the Universe with Cosmological Constant International Journal of Reciprocal Symmetry and Theoretical Physics, 1, 43-68.

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Published

2014-12-31

Issue

Vol. 3 No. 2 (2014): July-December

Section

Articles

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How to Cite

Das, S. . (2014). Congruence Properties of Andrews’SPT- Function. ABC Journal of Advanced Research, 3(2), 115-124. https://doi.org/10.18034/abcjar.v3i2.37
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