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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Author(s): Krishnaswami Alladi
Journal: Trans. Amer. Math. Soc. 349 (1997), 2721-2735.
MSC (1991): Primary 05A17, 05A19; Secondary 11P83
MathSciNet review: 1422593
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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.

References:

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 $^{\text {th}}$ 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., 66 AMS, Providence, 1986. MR 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
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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: 10.1090/S0002-9947-97-01944-2
PII: S 0002-9947(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.
Copyright of article: Copyright 1997, American Mathematical Society

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