On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture

Abstract

Let cR={r+jm:j≥0} and cS={s+jm:j≥0} be disjoint arithmetic progressions of positive integers, and choose a partition of n uniformly at random. We determine the limiting probability that the number of parts in cR is larger than the number of parts in cS. More precisely, if a=r/m and b=s/m, then this probability converges to I_1/2(a,b), the regularized incomplete beta function. We prove the stronger distributional statement that, after multiplication by the partition saddle-point parameter, the difference of the two numbers of parts converges to m⁻¹(log G_b−log G_a), where G_a and G_b are independent gamma variables. The proof uses the Boltzmann conditioning device for random partitions, a q-gamma evaluation of the unconditioned characteristic function, and a uniform Fourier estimate showing that conditioning on the total size does not change the limit. We also evaluate the finite-truncation limit studied by Li and Zhao. The two evaluations coincide, proving their conjecture on the commutation of the truncation and large-size limits.

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BibTeX RIS

BibTeX

@misc{li2026arithmeticprogression,
  title = {{On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture}},
  author = {Jiyou Li and Sicheng Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30700},
  eprint = {2610.30700},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30700},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On Arithmetic-Progression Biases in Random Partitions: Limiting Laws and the Li-Zhao Conjecture
AU  - Jiyou Li
AU  - Sicheng Zhao
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.30700
UR  - https://arxiv.org/abs/2610.30700
AB  - Let cR={r+jm:j≥0} and cS={s+jm:j≥0} be disjoint arithmetic progressions of positive integers, and choose a partition of n uniformly at random. We determine the limiting probability that the number of parts in cR is larger than the number of parts in cS. More precisely, if a=r/m and b=s/m, then this probability converges to I_1/2(a,b), the regularized incomplete beta function. We prove the stronger distributional statement that, after multiplication by the partition saddle-point parameter, the difference of the two numbers of parts converges to m^−1(log G_b−log G_a), where G_a and G_b are independent gamma variables. The proof uses the Boltzmann conditioning device for random partitions, a q-gamma evaluation of the unconditioned characteristic function, and a uniform Fourier estimate showing that conditioning on the total size does not change the limit. We also evaluate the finite-truncation limit studied by Li and Zhao. The two evaluations coincide, proving their conjecture on the commutation of the truncation and large-size limits.
ER  -

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