<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
Dyson's New Symmetry and Generalized Rogers-Ramanujan Identities
Cilanne Boulet
3813 boul. Lasalle, Verdun, QC H4G 1Z7, Canada
Annals of Combinatorics 14 (2) pp.159-191 Summer, 2010
AMS Subject Classification: 05A17, 11P81
We present a generalization, which we call (k, m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries bijectively. Using the two symmetries we give a new combinatorial proof of generalized Rogers-Ramanujan identities. We also describe the relationship between (k, m)-rank and Garvan's k-rank.
Keywords: Rogers-Ramanujan identity, Schur's identity, successive Durfee squares, Dyson's rank, bijection, integer partition


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

2. Andrews, G.E.: An analytic generalization of the Rogers-Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci. USA 71, 4082––4085 (1974)

3. Andrews, G.E.: The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading (1976)

4. Andrews, G.E.: Partitions and Durfee dissection. Amer. J. Math. 101, 735––742 (1979)

5. 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, 341––350 (1987)

6. Berkovich, A., Garvan, F.G.: Some observations on Dyson's new symmetries of partitions. J. Combin. Theory Ser. A 100, 61––93 (2002)

7. Boulet, C.E.: Partition identity bijections related to sign-balance and rank. Massachusetts Institute of Technology, Cambridge (2005)

8. Boulet, C.E., Pak, I.: A new combinatorial proof of the Rogers-Ramanujan and Schur identities. J. Combin. Theory Ser. A 113, 1019––1030 (2006)

9. Bressoud, D.M.: A generalization of the Rogers-Ramanujan identities for all moduli. J. Combin. Theory Ser. A 27, 64––68 (1979)

10. Bressoud, D.M.: An analytic generalization of the Rogers-Ramanujan identities with interpretation. Quart. J. Math. Oxford Ser. (2) 31, 385––399 (1980)

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

12. Bressoud, D.M.: An easy proof of the Rogers-Ramanujan identities. J. Number Theory 16, 235––241 (1983)

13. Bressoud, D.M., Zeilberger, D.: A short Rogers-Ramanujan bijection. Discrete Math. 38, 313––315 (1982)

14. Bressoud, D.M., Zeilberger, D.: Generalized Rogers-Ramanujan bijections. Adv. Math. 78, 42––75 (1989)

15. Dyson, F.J.: Some guesses in the theory of partitions. Eureka 8, 10––15 (1944)

16. Dyson, F.J.: A new symmetry of partitions. J. Combin. Theory 7, 61––65 (1969)

17. Dyson, F.J.: A walk through Ramanujan's garden, In: Ramanujan revisited, pp. 7–28. Academic Press, Boston (1988)

18. Garrett, K., Ismail, M.E., Stanton, D.: Variants of the Rogers-Ramanujan identities. Adv. in Appl. Math. 23, 274––299 (1999)

19. Garvan, F.G.: Generalizations of Dyson's rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84, 343––359 (1994)

20. Pak, I.: On Fine's partition theorems, Dyson, Andrews, and missed opportunities. Math. Intelligencer 25, 10––16 (2003)

21. Pak, I.: Partition bijections, a survey. Ramanujan J. 12, 5––75 (2006)

22. Rogers, L.J.: Second memoir on the expansion of certain infinite products. Proc. London Math. Soc. 25, 318––343 (1894)

23. Schur, I.: Ein Beitrag zur Additiven Zhalentheorie und zur Theorie der Kettenbrüche. S.-B. Preuss. Akad. Wiss. Phys. Math. Klasse 302––321 (1917)

24. Sylvester, J.J., Franklin, F.: A constructive theory of partitions, arranged in three acts, an interact and an exodion. Amer. J. Math. 5, 251––330 (1882)

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