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PreprintCC BY-NC-SA 4.0arXiv

The e-positivity of rowDivk(eμ)

  • Per Alexandersson
  • Lilan Dai
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Abstract

We prove that the partition division operator rowDivk preserves e-positivity, resolving a conjecture of Alexandersson and Dai. For μ⊢ kn and ν⊢ n, we express the coefficient of eν in rowDivk(eμ) as a normalized sum of positive rational weights over closed walks on a finite directed graph of k-restricted partitions. The proof combines the Pieri rule and a root of unity factorization for Macdonald polynomials. As a corollary, we recover the h-positivity of the symmetric functions An,k of Amdeberhan, Shareshian, and Stanley.

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Cite

BibTeXRIS
BibTeX
@misc{alexandersson2026epositivity,
  title = {{The e-positivity of $\mathrmrowDiv_k(e_μ)$}},
  author = {Per Alexandersson and Lilan Dai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.26303},
  eprint = {2610.26303},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.26303},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - The e-positivity of rowDiv_k(e_μ)
AU  - Per Alexandersson
AU  - Lilan Dai
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.26303
UR  - https://arxiv.org/abs/2610.26303
AB  - We prove that the partition division operator rowDiv_k preserves e-positivity, resolving a conjecture of Alexandersson and Dai. For μ⊢ kn and ν⊢ n, we express the coefficient of e_ν in rowDiv_k(e_μ) as a normalized sum of positive rational weights over closed walks on a finite directed graph of k-restricted partitions. The proof combines the Pieri rule and a root of unity factorization for Macdonald polynomials. As a corollary, we recover the h-positivity of the symmetric functions A_n,k of Amdeberhan, Shareshian, and Stanley.
ER  -