Partition identities involving gaps and weights

Author:
Krishnaswami Alladi

Journal:
Trans. Amer. Math. Soc. **349** (1997), 5001-5019

MSC (1991):
Primary 11P83, 11P81; Secondary 05A19

MathSciNet review:
1401759

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers-Ramanujan 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.

**1.**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)*.**3.**Krishnaswami Alladi, George Andrews, and Basil Gordon,*Refinements and generalizations of Capparelli’s conjecture on partitions*, J. Algebra**174**(1995), no. 2, 636–658. MR**1334229**, 10.1006/jabr.1995.1144**4.**Krishnaswami Alladi and Basil Gordon,*Partition identities and a continued fraction of Ramanujan*, J. Combin. Theory Ser. A**63**(1993), no. 2, 275–300. MR**1223685**, 10.1016/0097-3165(93)90061-C**5.**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****6.**George E. Andrews,*Partitions and Durfee dissection*, Amer. J. Math.**101**(1979), no. 3, 735–742. MR**533197**, 10.2307/2373804**7.**George E. Andrews,*Partitions: yesterday and today*, New Zealand Mathematical Society, Wellington, 1979. With a foreword by J. C. Turner. MR**557539****8.**Douglas Bowman,*Partitions with numbers in their gaps*, Acta Arith.**74**(1996), no. 2, 97–105. MR**1373701****9.**D. M. Bressoud,*On a partition theorem of Göllnitz*, J. Reine Angew. Math.**305**(1979), 215–217. MR**518863**, 10.1515/crll.1979.305.215**10.**Frank G. Garvan,*Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions*, Manuscripta Math.**84**(1994), no. 3-4, 343–359. MR**1291125**, 10.1007/BF02567461**11.**H. Göllnitz,*Partitionen mit Differenzenbedingungen*, J. Reine Angew. Math.**225**(1967), 154–190 (German). MR**0211973****12.**Basil Gordon,*A combinatorial generalization of the Rogers-Ramanujan identities*, Amer. J. Math.**83**(1961), 393–399. MR**0123484****13.**B. Gordon and K. Ono,*Divisibility of certain partition functions by powers of primes*, The Ramanujan Journal**1**(1997), 25-34.**14.**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.)****15.**Jean-Pierre Serre,*Divisibilité des coefficients des formes modulaires de poids entier*, C. R. Acad. Sci. Paris Sér. A**279**(1974), 679–682 (French). MR**0382172****16.**J. J. Sylvester,*A constructive theory of partitions arranged in three Acts, an Interact and an Exodion*, Amer. J. Math.**5**(1882), 251-330.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
11P83,
11P81,
05A19

Retrieve articles in all journals with MSC (1991): 11P83, 11P81, 05A19

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/S0002-9947-97-01831-X

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