A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications

Author:
Krishnaswami Alladi

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2721-2735

MSC (1991):
Primary 05A17, 05A19; Secondary 11P83

MathSciNet review:
1422593

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.

**1.**K. Alladi,*Partition identities involving gaps and weights*, Trans. Amer. Math. Soc. (to appear).**2.**K. Alladi, and G. E. Andrews,*A new key identity for Göllnitz' (big) partition theorem*in Proc. 10 Anniv. Conf. Ramanujan Math. Soc., Contemp. Math. (to appear).**3.**Krishnaswami Alladi, George E. Andrews, and Basil Gordon,*Generalizations and refinements of a partition theorem of Göllnitz*, J. Reine Angew. Math.**460**(1995), 165–188. MR**1316576****4.**Krishnaswami Alladi and Basil Gordon,*Generalizations of Schur’s partition theorem*, Manuscripta Math.**79**(1993), no. 2, 113–126. MR**1216769**, 10.1007/BF02568332**5.**George E. Andrews,*On a partition theorem of Göllnitz and related formulae*, J. Reine Angew. Math.**236**(1969), 37–42. MR**0248101****6.**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****7.**George E. Andrews,*𝑞-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra*, CBMS Regional Conference Series in Mathematics, vol. 66, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR**858826****8.**David M. Bressoud,*A combinatorial proof of Schur’s 1926 partition theorem*, Proc. Amer. Math. Soc.**79**(1980), no. 2, 338–340. MR**565367**, 10.1090/S0002-9939-1980-0565367-X**9.**H. Göllnitz,*Partitionen mit Differenzenbedingungen*, J. Reine Angew. Math.**225**(1967), 154–190 (German). MR**0211973****10.**Issai Schur,*Gesammelte Abhandlungen. Band II*, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR**0462892****11.**J. J. Sylvester,*A constructive theory of partitions arranged in three Acts, an Interact and an Exodion*, Amer. J. Math.**5**(1882), 251-330.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
05A17,
05A19,
11P83

Retrieve articles in all journals with MSC (1991): 05A17, 05A19, 11P83

Additional Information

**Krishnaswami Alladi**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611

Email:
alladi@math.ufl.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-97-01944-2

Keywords:
Partitions,
G\"{o}llnitz' theorem,
distinct parts,
weighted words,
Sylvester's identity,
sliding operation

Received by editor(s):
September 1, 1995

Additional Notes:
Research supported in part by National Science Foundation grant DMS 9400191.

Article copyright:
© Copyright 1997
American Mathematical Society