The combinatorics of 𝑘-marked Durfee symbols

Ji, Kathy

doi:10.1090/S0002-9947-2010-05136-0

The combinatorics of $k$-marked Durfee symbols
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by Kathy Qing Ji

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

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

Published electronically: September 21, 2010

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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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