Skip to main content
SpringerLink
Log in

Singular Overpartitions and Partitions with Prescribed Hook Differences

  • Published:
Annals of Combinatorics Aims and scope Submit manuscript

Abstract

Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (K, i)-singular overpartitions for integers K, i with \( 1\le i<K/2\). The definition of such singular overpartitions requires successive ranks, parity blocks and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in 1987. In this paper, employing hook differences, we generalize parity blocks. Using this combinatorial concept, we define \((K,i,\alpha , \beta )\)-singular overpartitions for positive integers \(\alpha , \beta \) with \(\alpha +\beta <K\), and then we show some connections between such singular overpartitions and ordinary partitions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Andrews, G.E.: Sieves for theorems of Euler, Rogers, and Ramanujan. In: Gioia, A.A., Goldsmith, D.L. (eds.) The Theory of Arithmetic Functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1971), pp. 1–20. Lecture Notes in Math., Vol. 251, Springer, Berlin (1972)

    Google Scholar 

  2. Andrews, G.E.: Sieves in the theory of partitions. Amer. J. Math. 94, 1214–1230 (1972)

    Article  MathSciNet  Google Scholar 

  3. Andrews, G.E.: The Theory of Partitions. Encyclopedia of Mathematics and its Applications, Vol. 2. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam. (1976); reissued: Cambridge University Press, Cambridge (1998)

  4. Andrews, G.E.: Generalized Frobenius Partitions. Mem. Amer. Math. Soc. 49(301), iv+44 (1984)

  5. Andrews, G.E.: Singular overpartitions. Int. J. Number Theory 11(5), 1523–1533 (2015)

    Article  MathSciNet  Google Scholar 

  6. Andrews, G.E., Baxter, R.J., Bressoud, D.M., Burge, W.H., Forrester, P.J., Viennot, G.: Partitions with prescribed hook differences. European J. Combin. 8(4), 341–350 (1987)

    Article  MathSciNet  Google Scholar 

  7. Bressoud, D.M.: Extension of the partition sieve. J. Number Theory 12(1) 87–100 (1980)

    Article  MathSciNet  Google Scholar 

  8. Bressoud, D.M.: The Borwein conjecture and partitions with prescribed hook differences. Electron. J. Combin. 3(2), #R 4 (1996)

  9. Burge, W.H.: A correspondence between partitions related to generalizations of the Rogers-Ramanujan identities. Discrete Math. 34(1), 9–15 (1981)

    Article  MathSciNet  Google Scholar 

  10. Burge, W.H.: A three-way correspondence between partitions. European J. Combin. 3(3), 195–213 (1982)

    Article  MathSciNet  Google Scholar 

  11. Burge, W.H.: Combinatorial interpretations of some identities of the Rogers-Ramanujan type. I.B.M. Research Report, RC 9329 (#41101) (1982)

  12. Corteel, S., Lovejoy, J.: Overpartitions. Trans. Amer. Math. Soc. 356(4), 1623–1635 (2004)

    Article  MathSciNet  Google Scholar 

  13. Dyson, F.J.: A new symmetry of partitions. J. Combinatorial Theory 7, 56–61 (1969)

    Article  MathSciNet  Google Scholar 

  14. Gessel, I.M., Krattenthaler, C.: Cylindric partitions. Trans. Amer. Math. Soc. 349(2), 429–479 (1997)

    Article  MathSciNet  Google Scholar 

  15. Seo, S., Yee, A.J.: Overpartitions and singular overpartitions. In: Andrews, G.E., Garvan, F. (eds.) Analytic Number Theory, Modular Forms and \(q\)-Hypergeometric Series, pp. 693–711. Springer Proc. Math. Stat., 221, Springer, Cham (2017)

  16. Seo, S., Yee, A.J.: Enumeration of partitions with prescribed successive rank parity blocks. J. Combin. Theory Ser. A 158, 12–35 (2018)

    Article  MathSciNet  Google Scholar 

  17. Wright, E.M.: An enumerative proof of an identity of Jacobi. J. London Math. Soc. 40, 55–57 (1965)

    Article  MathSciNet  Google Scholar 

  18. Yee, A.J.: Combinatorial proofs of generating function identities for F-partitions. J. Combin. Theory Ser. A 102(1), 217–228 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics Education, Kangwon National University, Chuncheon, 24341, Gangwon, Republic of Korea

    Seunghyun Seo

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Ae Ja Yee

Authors
  1. Seunghyun Seo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ae Ja Yee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ae Ja Yee.

Additional information

Dedicated to George Andrews for his 80th birthday

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

S. Seo was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (120180215). A. J. Yee was partially supported by a grant (#280903) from the Simons Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Seo, S., Yee, A.J. Singular Overpartitions and Partitions with Prescribed Hook Differences. Ann. Comb. 23, 1039–1072 (2019). https://doi.org/10.1007/s00026-019-00466-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00026-019-00466-3

Keywords

Mathematics Subject Classification

Search

Navigation