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

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

MathSciNet review:
1422593

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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.**K. Alladi, G. E. Andrews and B. Gordon,*Generalizations and refinements of a partition theorem of Göllnitz*, J. Reine Angew. Math.**460**(1995), 165-188. MR**96c:11119****4.**K. Alladi and B. Gordon,*Generalizations of Schur's partition theorem*, Manus. Math.**79**(1993), 113-126. MR**94c:11099****5.**G. E. Andrews,*A partition theorem of Göllnitz and related formula*, J. Reine Angew. Math.,**236**(1969), 18-24. MR**40:1355****6.**G. E. Andrews,*The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2*, Addison-Wesley, Reading, MA (1976). MR**58:27738****7.**G. E. Andrews,*Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, CBMS Regional Conf. Ser. in Math.,*, 1986. MR**66**AMS, Providence**88b:11063****8.**D. M. Bressoud,*A combinatorial proof of Schur's 1926 partition theorem*, Proc. Amer. Math. Soc.,**79**(1980), 333-340. MR**81f:05017****9.**H. Göllnitz,*Partitionen mit Differenzenbedingungen*, J. Reine Angew. Math.,**225**(1967), 154-190. MR**35:2848****10.**I. Schur,*Zur additiven Zahlentheorie, Gesammelte Abhandlungen, Vol. 2*, Springer Verlag, Berlin (1973), 43-50. MR**57:2858b****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.

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Additional Information

**Krishnaswami Alladi**

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

Email:
alladi@math.ufl.edu

DOI:
https://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