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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [1]
Krishnaswami
Alladi and Basil
Gordon, Generalizations of Schur’s partition theorem,
Manuscripta Math. 79 (1993), no. 2, 113–126. MR 1216769
(94c:11099), http://dx.doi.org/10.1007/BF02568332
 [2]
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
(96c:11119)
 [3]
, Refinements and generalizations of Capparelli's conjecture on partitions, J. Algebra (to appear).
 [4]
George
E. Andrews, The theory of partitions, AddisonWesley
Publishing Co., Reading, Mass.LondonAmsterdam, 1976. Encyclopedia of
Mathematics and its Applications, Vol. 2. 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]
George
E. Andrews, The use of computers in search of identities of the
RogersRamanujan type, Computers in number theory (Proc. Sci. Res.
Council Atlas Sympos. No. 2, Oxford, 1969), Academic Press, London, 1971,
pp. 377–387. 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), 215–217. MR 518863
(80a:10027), http://dx.doi.org/10.1515/crll.1979.305.215
 [11]
David
M. Bressoud, A combinatorial proof of Schur’s
1926 partition theorem, Proc. Amer. Math.
Soc. 79 (1980), no. 2, 338–340. MR 565367
(81f:05017), http://dx.doi.org/10.1090/S0002993919800565367X
 [12]
Stefano
Capparelli, On some representations of twisted affine Lie algebras
and combinatorial identities, J. Algebra 154 (1993),
no. 2, 335–355. MR 1206124
(94d:17031), http://dx.doi.org/10.1006/jabr.1993.1017
 [13]
H.
Göllnitz, Partitionen mit Differenzenbedingungen, J.
Reine Angew. Math. 225 (1967), 154–190 (German). MR 0211973
(35 #2848)
 [14]
I. J. Schur, Zur additiven Zahlentheorie, Gessammelte Abhandlungen, vol. 2, SpringerVerlag, Berlin, 1973, pp. 4350.
 [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.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
11P83,
05A17,
05A19,
11P81
Retrieve articles in all journals
with MSC:
11P83,
05A17,
05A19,
11P81
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719951297520X
PII:
S 00029947(1995)1297520X
Article copyright:
© Copyright 1995 American Mathematical Society
