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Counting self-conjugate \((s,s+1,s+2)\)-core partitions

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Abstract

We are concerned with counting self-conjugate \((s,s+1,s+2)\)-core partitions. A Motzkin path of length n is a path from (0, 0) to (n, 0) which stays weakly above the x-axis and consists of the up \(U=(1,1)\), down \(D=(1,-1)\), and flat \(F=(1,0)\) steps. We say that a Motzkin path of length n is symmetric if its reflection about the line \(x=n/2\) is itself. In this paper, we show that the number of self-conjugate \((s,s+1,s+2)\)-cores is equal to the number of symmetric Motzkin paths of length s, and give a closed formula for this number.

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Acknowledgements

The authors thank an anonymous referee for his/her careful comments.

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Authors and Affiliations

  1. Institute of Mathematical Sciences, Ewha Womans University, Seoul, Republic of Korea

    Hyunsoo Cho

  2. Department of Mathematics, Ajou University, Suwon, Republic of Korea

    JiSun Huh

  3. Department of Mathematics, Yonsei University, Seoul, Republic of Korea

    Jaebum Sohn

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  1. Hyunsoo Cho
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Correspondence to JiSun Huh.

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Hyunsoo Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1A6A1A11051177). JiSun Huh was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01008524). Jaebum Sohn was supported by the National Research Foundation of Korea (NRF) NRF-2017R1A2B4009501.

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Cho, H., Huh, J. & Sohn, J. Counting self-conjugate \((s,s+1,s+2)\)-core partitions. Ramanujan J 55, 163–174 (2021). https://doi.org/10.1007/s11139-020-00300-y

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  • DOI: https://doi.org/10.1007/s11139-020-00300-y

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