Partition identities involving gaps and weights
Author:
Krishnaswami Alladi
Journal:
Trans. Amer. Math. Soc. 349 (1997), 50015019
MSC (1991):
Primary 11P83, 11P81; Secondary 05A19
MathSciNet review:
1401759
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Abstract: We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, RogersRamanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.
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 K. Alladi, A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications, Trans. Amer. Math. Soc. (to appear).
 2.
 K. Alladi, On the divisibility of certain partition functions by small powers of 2 (in preparation).
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 K. Alladi, G. E. Andrews and B. Gordon, Refinements and generalizations of Capparelli's conjecture on partitions, J. Algebra 174 (1995), 636658. MR 96b:11136
 4.
 K. Alladi and B. Gordon, Partition identities and a continued fraction of Ramanujan, J. Comb. Th. Ser. A 63 (1993), 275300. MR 94f:11108
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 G. E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, AddisonWesley, Reading (1976). MR 58:27738
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 G. E. Andrews, Partitions and Durfee dissection, Amer. J. Math. 10 (1979), 735742. MR 80h:10020
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 G. E. Andrews, Partitions: Yesterday and Today, New Zealand Math. Soc., Wellington (1979). MR 81b:01015
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 D. Bowman, Partitions with numbers in their gaps, Acta Arith. 74 (1996), 97105. MR 96m:11090
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 D. M. Bressoud, On a partition theorem of Göllnitz, J. Reine Angew. Math. 305 (1979), 215217. MR 80a:10027
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 F. Garvan, Generalizations of Dyson's rank and nonRogersRamanujan partitions, Manus. Math 84 (1994), 343359. MR 95e:11115
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 B. Gordon and K. Ono, Divisibility of certain partition functions by powers of primes, The Ramanujan Journal 1 (1997), 2534.
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 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 4th ed., Clarendon Press, Oxford (1960). MR 81i:10002 (5th ed.)
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 J. P. Serre, Divisibilité des coefficients des formes modulaires de poids entiers, C. R. Acad. Sci. Paris A 279 (1974), 679682. MR 52:3060
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 J. J. Sylvester, A constructive theory of partitions arranged in three Acts, an Interact and an Exodion, Amer. J. Math. 5 (1882), 251330.
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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/S000299479701831X
PII:
S 00029947(97)01831X
Keywords:
Partitions,
weights,
gaps,
Durfee squares,
powers of 2
Received by editor(s):
October 10, 1995
Received by editor(s) in revised form:
June 3, 1996
Additional Notes:
Research supported in part by the National Science Foundation Grant DMS 9400191.
Article copyright:
© Copyright 1997
American Mathematical Society
