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The combinatorics of $ k$-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 $ k$-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 $ k$-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on $ k$-marked Durfee symbols given by Andrews could follow from this identity. In a similar manner, we also prove the identities due to Andrews on $ k$-marked odd Durfee symbols combinatorially, which resemble ordinary $ k$-marked Durfee symbols with a modified subscript and with odd numbers as entries.

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Additional Information

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

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.

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