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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On signed multiplicities of Schur expansions surrounding Petrie symmetric functions
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by Yen-Jen Cheng, Meng-Chien Chou, Sen-Peng Eu, Tung-Shan Fu and Jyun-Cheng Yao
Proc. Amer. Math. Soc. 151 (2023), 1839-1854
DOI: https://doi.org/10.1090/proc/16263
Published electronically: February 2, 2023

Abstract:

For $k\ge 1$, the homogeneous symmetric functions $G(k,m)$ of degree $m$ defined by $\sum _{m\ge 0} G(k,m) z^m=\prod _{i\ge 1} \big (1+x_iz+x^2_iz^2+\cdots +x^{k-1}_iz^{k-1}\big )$ are called Petrie symmetric functions. As derived by Grinberg and Fu–Mei independently, the expansion of $G(k,m)$ in the basis of Schur functions $s_{\lambda }$ turns out to be signed multiplicity free, i.e., the coefficients are $-1$, $0$ and $1$. In this paper we give a combinatorial interpretation of the coefficient of $s_{\lambda }$ in terms of the $k$-core of $\lambda$ and a sequence of rim hooks of size $k$ removed from $\lambda$. We further study the product of $G(k,m)$ with a power sum symmetric function $p_n$. For all $n\ge 1$, we give necessary and sufficient conditions on the parameters $k$ and $m$ in order for the expansion of $G(k,m)\cdot p_n$ in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case $n=2$.
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Bibliographic Information
  • Yen-Jen Cheng
  • Affiliation: Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan, Republc of China
  • MR Author ID: 1261410
  • Email: yjc7755@nycu.edu.tw
  • Meng-Chien Chou
  • Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116325, Taiwan, Republc of China
  • Email: aoliver466@gmail.com
  • Sen-Peng Eu
  • Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116325, Taiwan, Republc of China
  • MR Author ID: 705346
  • ORCID: 0000-0002-2645-2625
  • Email: speu@math.ntnu.edu.tw
  • Tung-Shan Fu
  • Affiliation: Department of Applied Mathematics, National Pingtung University, Pingtung 900391, Taiwan, Republc of China
  • MR Author ID: 350587
  • Email: tsfu@mail.nptu.edu.tw
  • Jyun-Cheng Yao
  • Affiliation: Department of Mathematics, National Taiwan Normal University, Taipei 116325, Taiwan, Republc of China
  • Email: 0955526287er@gmail.com
  • Received by editor(s): June 28, 2022
  • Received by editor(s) in revised form: July 21, 2022, August 7, 2022, August 17, 2022, August 18, 2022, and August 20, 2022
  • Published electronically: February 2, 2023
  • Additional Notes: This research was supported in part by Ministry of Science and Technology (MOST), Taiwan, grants 110-2115-M-003-011-MY3 (the third author), 111-2115-M-153-004-MY2 (the fourth author), and MOST postdoctoral fellowship 111-2811-M-A49-537-MY2 (the first author).
  • Communicated by: Isabella Novik
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 1839-1854
  • MSC (2020): Primary 05E05; Secondary 05A17
  • DOI: https://doi.org/10.1090/proc/16263
  • MathSciNet review: 4556182