On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo and generalizations

Authors:
Alexander Berkovich and Frank G. Garvan

Journal:
Trans. Amer. Math. Soc. **358** (2006), 703-726

MSC (2000):
Primary 11P81, 11P83; Secondary 05A17, 05A19

DOI:
https://doi.org/10.1090/S0002-9947-05-03751-7

Published electronically:
March 10, 2005

MathSciNet review:
2177037

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic

where denotes the number of odd parts of the partition and is the conjugate of . In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod :

where () denotes the number of partitions of with and is the number of unrestricted partitions of . Andrews asked for a partition statistic that would divide the partitions enumerated by () into five equinumerous classes.

In this paper we discuss three such statistics: the ST-crank, the -quotient-rank and the -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo . Finally, we discuss some new formulas for partitions that are -cores and discuss an intriguing relation between -cores and the Andrews-Garvan crank.

**1.**G. E. Andrews,*On a partition function of Richard Stanley*, to appear in the Electronic Journal of Combinatorics volume in honor of Richard Stanley.**2.**G. E. Andrews and F. G. Garvan,*Dyson's crank of a partition*, Bull. Amer. Math. Soc. (N.S.)**18**(1988), 167-171. MR**0929094 (89b:11079)****3.**A. O. L. Atkin and P. Swinnerton-Dyer,*Some properties of partitions*, Proc. London Math. Soc.**4**(1954), 84-106. MR**0060535 (15:685d)****4.**C. E. Boulet,*A four-parameter partition identity*, preprint.**5.**F. J. Dyson,*Some guesses in the theory of partitions*, Eureka (Cambridge)**8**(1944), 10-15.**6.**F. G. Garvan,*New combinatorial interpretations of Ramanujan's partition congruences mod and*, Trans. Amer. Math. Soc.**305**(1988), 47-77. MR**0920146 (89b:11081)****7.**F. G. Garvan,*The crank of partitions mod and*, Trans. Amer. Math. Soc.**322**(1990), 79-94. MR**1012520 (91b:11109)****8.**F. G. Garvan,*More cranks and -cores*, Bull. Austral. Math. Soc.**63**(2001), 379-391. MR**1834941 (2002f:11140)****9.**F. Garvan, D. Kim and D. Stanton,*Cranks and -cores*, Invent. Math.**101**(1990), 1-17. MR**1055707 (91h:11106)****10.**M. Hirschhorn, F. Garvan and J. Borwein,*Cubic analogues of the Jacobian theta function*, Canad. J. Math.**45**(1993), 673-694. MR**1227653 (94m:33011)****11.**G. James and A. Kerber,*The Representation Theory of the Symmetric Group*, Addison-Wesley, Reading, MA, 1981. MR**0644144 (83k:20003)****12.**D. E. Littlewood,*Modular representations of symmetric groups*, Proc. Roy. Soc. London. Ser. A.**209**(1951), 333-353. MR**0049896 (14:243b)****13.**A. V. Sills,*A combinatorial proof of a partition identity of Andrews and Stanley*, preprint.**14.**R. P. Stanley,*Some remarks on sign-balanced and maj-balanced posets*, preprint.**15.**A. J. Yee,*On partition functions of Andrews and Stanley*, preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11P81,
11P83,
05A17,
05A19

Retrieve articles in all journals with MSC (2000): 11P81, 11P83, 05A17, 05A19

Additional Information

**Alexander Berkovich**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Email:
alexb@math.ufl.edu

**Frank G. Garvan**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105

Email:
frank@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03751-7

Keywords:
Partitions,
$t$-cores,
ranks,
cranks,
Stanley's statistic,
Ramanujan's congruences

Received by editor(s):
January 12, 2004

Received by editor(s) in revised form:
February 24, 2004

Published electronically:
March 10, 2005

Article copyright:
© Copyright 2005
American Mathematical Society