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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, G.E.: The Theory of Partitions. Encycl. Math. Appl., vol. 2. Addison-Wesley, Reading (1976) (Reissued: Cambridge University Press, Cambridge (1985))
Andrews, G.E.: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS Regional Conference Series in Mathematics, vol. 66. AMS, Providence (1986)
Andrews, G.E.: Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks. Invent. Math. 169, 37–73 (2007). doi:10.1007/s00222-007-0043-4
Andrews, G.E.: The number of smallest parts in the partitions of n. J. Reine Angew. Math. 624, 133–142 (2008). doi:10.1515/CRELLE.2008.083
Andrews, G.E., Dyson, F.J., Hickerson, D.: Partitions and indefinite quadratic forms. Invent. Math. 91, 391–407 (1988). doi:10.1007/BF01388778
Andrews, G.E., Garvan, F.G.: Dyson’s crank of a partition. Bull., New Ser., Am. Math. Soc. 18, 167–171 (1988). doi:10.1090/S0273-0979-1988-15637-6
Andrews, G.E., Garvan, F.G., Liang, J.: Self-conjugate vector partitions and the parity of the spt-function. Preprint
Atkin, A.O.L., Swinnerton-Dyer, P.: Some properties of partitions. Proc. Lond. Math. Soc. 4(3), 84–106 (1954). doi:10.1112/plms/s3-4.1.84
Berndt, B.C., Chan, H.H., Chan, S.H., Liaw, W.-C.: Cranks—really, the final problem. Ramanujan J. 23, 3–15 (2010). doi:10.1007/s11139-008-9143-z
Dyson, F.J.: Some guesses in the theory of partitions. Eureka (Camb.) 8, 10–15 (1944)
Folsom, A., Ono, K.: The spt-function of Andrews. Proc. Natl. Acad. Sci. USA 105, 20152–20156 (2008). http://mathcs.emory.edu/~ono/publications-cv/pdfs/111.pdf
Garvan, F.G.: New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11. Trans. Am. Math. Soc. 305, 47–77 (1988). doi:10.2307/2001040
Garvan, F.G.: Combinatorial interpretations of Ramanujan’s partition congruences. In: Ramanujan Revisited: Proc. of the Centenary Conference, Univ. of Illinois at Urbana-Champaign, 1–5 June 1987. Academic Press, San Diego (1988). http://www.math.ufl.edu/~fgarvan/papers/urbana.pdf
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encycl. Math. Appl. Cambridge University Press, Cambridge (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported in part by NSA Grant H9820-12-1-0205.
The second author was supported in part by NSA Grant H98230-09-1-0051.
The third author was supported by the Summer Research Experience for Rising Seniors (SRRS) program of the University of Florida with funding from the Howard Hughes Medical Institute the Science for Life Program.
Rights and permissions
About this article
Cite this article
Andrews, G.E., Garvan, F.G. & Liang, J. Combinatorial interpretations of congruences for the spt-function. Ramanujan J 29, 321–338 (2012). https://doi.org/10.1007/s11139-012-9369-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-012-9369-7
Keywords
- Spt-function
- Partitions
- Rank
- Crank
- Vector partitions
- Ramanujan’s Lost Notebook
- Congruences
- Basic hypergeometric series