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Average size of a self-conjugate $ (s,t)$-core partition


Authors: William Y. C. Chen, Harry H. Y. Huang and Larry X. W. Wang
Journal: Proc. Amer. Math. Soc. 144 (2016), 1391-1399
MSC (2010): Primary 05A17, 05A15
DOI: https://doi.org/10.1090/proc/12729
Published electronically: December 21, 2015
MathSciNet review: 3451218
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Abstract: Armstrong, Hanusa and Jones conjectured that if $ s,t$ are coprime integers, then the average size of an $ (s,t)$-core partition and the average size of a self-conjugate $ (s,t)$-core partition are both equal to $ \frac {(s+t+1)(s-1)(t-1)}{24}$. Stanley and Zanello showed that the average size of an $ (s,s+1)$-core partition equals $ \binom {s+1}{3}/2$. Based on a bijection of Ford, Mai and Sze between self-conjugate $ (s,t)$-core partitions and lattice paths in an $ \lfloor \frac {s}{2} \rfloor \times \lfloor \frac {t}{2}\rfloor $ rectangle, we obtain the average size of a self-conjugate $ (s,t)$-core partition as conjectured by Armstrong, Hanusa and Jones.


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Additional Information

William Y. C. Chen
Affiliation: Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China
Email: chenyc@tju.edu.cn

Harry H. Y. Huang
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: haoyangh@mail.nankai.edu.cn

Larry X. W. Wang
Affiliation: Center for Combinatorics, LPMC-TJKLC, Nankai University, Tianjin 300071, People’s Republic of China
Email: wsw82@nankai.edu.cn

DOI: https://doi.org/10.1090/proc/12729
Keywords: $(s,t)$-core partition, self-conjugate partition, lattice path
Received by editor(s): May 11, 2014
Received by editor(s) in revised form: January 1, 2015
Published electronically: December 21, 2015
Additional Notes: We wish to thank the referee for helpful suggestions. This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education and the National Science Foundation of China.
Communicated by: Patricia Hersh
Article copyright: © Copyright 2015 American Mathematical Society