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.
Bibliographical noteFunding Information:
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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All Science Journal Classification (ASJC) codes
- Algebra and Number Theory