Schur's partition theorem, companions, refinements and generalizations
Authors:
Krishnaswami Alladi and Basil Gordon
Journal:
Trans. Amer. Math. Soc. 347 (1995), 15911608
MSC:
Primary 11P83; Secondary 05A17, 05A19, 11P81
MathSciNet review:
1297520
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Abstract: Schur's partition theorem asserts the equality , where is the number of partitions of into distinct parts and is the number of partitions of into parts with minimal difference and no consecutive multiples of . Using a computer search Andrews found a companion result , where is the number of partitions of whose parts satisfy according as or . By means of a new technique called the method of weighted words, a combinatorial as well as a generating function proof of both these theorems are given simultaneously. It is shown that and are only two of six companion partition functions , all equal to . A three parameter refinement and generalization of these results is obtained.
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 [1]
 K. Alladi and B. Gordon, Generalizations of Schur's partition theorem, Manuscripta Math. 79 (1993), 113126. MR 1216769 (94c:11099)
 [2]
 K. Alladi, G. E. Andrews and B. Gordon, Generalizations and refinements of a partition theorem of Göllnitz, J.Reine Angew. Math. (to appear). MR 1316576 (96c:11119)
 [3]
 , Refinements and generalizations of Capparelli's conjecture on partitions, J. Algebra (to appear).
 [4]
 G. E. Andrews, The theory of partitions, Encyclopedia of Math., vol. 2, AddisonWesley, Reading, MA, 1976. MR 0557013 (58:27738)
 [5]
 , On Schur's second partition theorem, Glasgow Math. J. 9 (1967), 127132.
 [6]
 , A new generalization of Schur's second partition theorem, Acta Arith. 4 (1968), 429434.
 [7]
 , A general partition theorem with difference conditions, Amer. J. Math. 191 (1969), 1824.
 [8]
 , The use of computers in search of identities of the RogersRamanujan type, Computers in Number Theory (A. O. L. Atkin and B. J. Birch, eds.), Academic Press, New York, 1971, pp. 377387. MR 0316373 (47:4920)
 [9]
 , Schur's theorem, Capparelli's conjecture and trinomial coefficients, The Rademacher Legacy to Mathematics (Proc. Rademacher Centenary Conf, 1992), Contemp. Math., Amer. Math. Soc., Providence, RI, 1994, pp. 141154.
 [10]
 D. M. Bressoud, On a partition theorem of Göllnitz, J. Reine Angew. Math. 305 (1979), 215217. MR 518863 (80a:10027)
 [11]
 , A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer. Math. Soc. 79 (1980), 338340. MR 565367 (81f:05017)
 [12]
 S. Capparelli, On some representations of twisted affine Lie algebras and combinatorial identities, J. Algebra 154 (1993), 335355. MR 1206124 (94d:17031)
 [13]
 H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154190. MR 0211973 (35:2848)
 [14]
 I. J. Schur, Zur additiven Zahlentheorie, Gessammelte Abhandlungen, vol. 2, SpringerVerlag, Berlin, 1973, pp. 4350.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719951297520X
PII:
S 00029947(1995)1297520X
Article copyright:
© Copyright 1995
American Mathematical Society
