The combinatorics of -marked Durfee symbols

Author:
Kathy Qing Ji

Journal:
Trans. Amer. Math. Soc. **363** (2011), 987-1005

MSC (2010):
Primary 11P81, 05A17, 05A19

Published electronically:
September 21, 2010

MathSciNet review:
2728593

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Abstract: George E. Andrews recently introduced -marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a partition identity, which gives a relation between -marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on -marked Durfee symbols given by Andrews could follow from this identity. In a similar manner, we also prove the identities due to Andrews on -marked odd Durfee symbols combinatorially, which resemble ordinary -marked Durfee symbols with a modified subscript and with odd numbers as entries.

**1.**George E. Andrews,*Problems and prospects for basic hypergeometric functions*, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 191–224. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. MR**0399528****2.**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****3.**George E. Andrews,*Plane partitions. III. The weak Macdonald conjecture*, Invent. Math.**53**(1979), no. 3, 193–225. MR**549398**, 10.1007/BF01389763**4.**George E. Andrews,*Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks*, Invent. Math.**169**(2007), no. 1, 37–73. MR**2308850**, 10.1007/s00222-007-0043-4**5.**A. O. L. Atkin and P. Swinnerton-Dyer,*Some properties of partitions*, Proc. London Math. Soc. (3)**4**(1954), 84–106. MR**0060535****6.**A. O. L. Atkin and F. G. Garvan,*Relations between the ranks and cranks of partitions*, Ramanujan J.**7**(2003), no. 1-3, 343–366. Rankin memorial issues. MR**2035811**, 10.1023/A:1026219901284**7.**Alexander Berkovich and Frank G. Garvan,*Some observations on Dyson’s new symmetries of partitions*, J. Combin. Theory Ser. A**100**(2002), no. 1, 61–93. MR**1932070**, 10.1006/jcta.2002.3281**8.**C. Boulet and K. Kursungoz,*Symmetry of -marked Durfee symbols*, Int. J. Number Theory, to appear.**9.**Kathrin Bringmann, Frank Garvan, and Karl Mahlburg,*Partition statistics and quasiharmonic Maass forms*, Int. Math. Res. Not. IMRN**1**(2009), Art. ID rnn124, 63–97. MR**2471296**, 10.1093/imrn/rnn124**10.**Kathrin Bringmann,*On the explicit construction of higher deformations of partition statistics*, Duke Math. J.**144**(2008), no. 2, 195–233. MR**2437679**, 10.1215/00127094-2008-035**11.**Kathrin Bringmann, Jeremy Lovejoy, and Robert Osburn,*Automorphic properties of generating functions for generalized rank moments and Durfee symbols*, Int. Math. Res. Not. IMRN**2**(2010), 238–260. MR**2581041**, 10.1093/imrn/rnp131**12.**F. J. Dyson,*Some guesses in the theory of partitions*, Eureka (Cambridge)**8**(1944) 10-15.**13.**Freeman J. Dyson,*A new symmetry of partitions*, J. Combinatorial Theory**7**(1969), 56–61. MR**0238711****14.**Ira M. Gessel and Guoce Xin,*A short proof of the Zeilberger-Bressoud 𝑞-Dyson theorem*, Proc. Amer. Math. Soc.**134**(2006), no. 8, 2179–2187 (electronic). MR**2213689**, 10.1090/S0002-9939-06-08224-4**15.**W. J. Keith,*Rank of Partitions and Degree Symbols*, The Pennsylvania State University, 2007.**16.**William J. Keith,*Distribution of the full rank in residue classes for odd moduli*, Discrete Math.**309**(2009), no. 16, 4960–4968. MR**2548896**, 10.1016/j.disc.2009.02.032**17.**K. Kursungoz,*Parity Considerations in Andrews-Gordon Identities and the -Marked Durfee Symbols*, The Pennsylvania State University, 2009.**18.**Guoce Xin,*A fast algorithm for MacMahon’s partition analysis*, Electron. J. Combin.**11**(2004), no. 1, Research Paper 58, 20. MR**2097324****19.**Guoce Xin,*A residue theorem for Malcev-Neumann series*, Adv. in Appl. Math.**35**(2005), no. 3, 271–293. MR**2164920**, 10.1016/j.aam.2005.02.004**20.**G. N. Watson,*The final problem*, J. Lond. Math. Soc.**11**(1936) 55-80.

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Additional Information

**Kathy Qing Ji**

Affiliation:
Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China

Email:
ji@nankai.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-2010-05136-0

Keywords:
Rank,
the moment of rank,
the symmetrized moment of rank,
Durfee symbols,
$k$-marked Durfee symbols,
odd Durfee symbols,
$k$-marked odd Durfee symbols

Received by editor(s):
June 16, 2008

Received by editor(s) in revised form:
June 8, 2009

Published electronically:
September 21, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.