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The (km,kn)-square paths theorem

  • Dun Qiu
  • Minhao Zhang
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Abstract

For positive integers k,m,n with gcd(m,n)=1, we prove a (km,kn)-square paths theorem: the symmetric function (−1)k−1[km]q/[k]qpk[−MXm,n]·1 is a weighted sum over (km,kn)-preference functions. This proves a corrected form of the conjecture of Iraci, Pagaria, Paolini, and Vanden Wyngaerd and generalizes the square paths theorem of Sergel. We encode preference functions by row strict tableaux and closed walks on a set of states, compare them with parking functions for each value of the touch statistic, and apply Mellit's compositional (km,kn)-shuffle theorem. Finally, we establish a fall Delta square theorem conjectured by Iraci, Pagaria, and Paolini.

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Cite

BibTeXRIS
BibTeX
@misc{qiu2026kmknsquare,
  title = {{The $(km,kn)$-square paths theorem}},
  author = {Dun Qiu and Minhao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16353},
  eprint = {2610.16353},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16353},
}
RIS
TY  - UNPB
TI  - The (km,kn)-square paths theorem
AU  - Dun Qiu
AU  - Minhao Zhang
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16353
UR  - https://arxiv.org/abs/2610.16353
AB  - For positive integers k,m,n with gcd(m,n)=1, we prove a (km,kn)-square paths theorem: the symmetric function (−1)^k−1[km]_q/[k]_qp_k[−MX^m,n]·1 is a weighted sum over (km,kn)-preference functions. This proves a corrected form of the conjecture of Iraci, Pagaria, Paolini, and Vanden Wyngaerd and generalizes the square paths theorem of Sergel. We encode preference functions by row strict tableaux and closed walks on a set of states, compare them with parking functions for each value of the touch statistic, and apply Mellit's compositional (km,kn)-shuffle theorem. Finally, we establish a fall Delta square theorem conjectured by Iraci, Pagaria, and Paolini.
ER  -