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On the number of even and odd strings along the overpartitions of n |
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| Authors: | Byungchan Kim Eunmi Kim Jeehyeon Seo |
| Affiliation: | 1. School of Liberal Arts, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowongu, Seoul, 139-743, Republic of Korea 2. Center for Applications of Mathematical Principles, National Institute for Mathematical Sciences, 70 Yuseong-daero 1689-gil, Yuseong-gu, Daejeon, 305-811, Republic of Korea 3. Department of Mathematics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul, 121-742, Republic of Korea |
| Abstract: | Recently, Andrews, Chan, Kim, and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture $$A_k (n) geq B_k (n)$$ holds for large enough positive integers n, where A k (n) (resp. B k (n)) is the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers. |
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