Zero-Run Spectra of the (3,2) Raney numbers Modulo Primes

Abstract

We study the zero-run structure of the (3,2)-Raney numbers modulo a prime p. For every prime p, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in 0≤ n<p^m. Interestingly, these results fall into three cases: p=2, p=3, and p≥5, and the behaviors in these three cases are very different. For p=2, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For p=3, we characterize exactly the nonzero terms and determine their residues. For p≥ 5, the zero runs are governed by a multiscale system of residue intervals modulo powers of p, from which both the record values and the complete zero-run spectrum are obtained.

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BibTeX RIS

BibTeX

@misc{eu2026zerorun,
  title = {{Zero-Run Spectra of the $(3,2)$ Raney numbers Modulo Primes}},
  author = {Sen-Peng Eu and Zai-Ting Huang and Louis Kao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25742},
  eprint = {2609.25742},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25742},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Zero-Run Spectra of the (3,2) Raney numbers Modulo Primes
AU  - Sen-Peng Eu
AU  - Zai-Ting Huang
AU  - Louis Kao
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25742
UR  - https://arxiv.org/abs/2609.25742
AB  - We study the zero-run structure of the (3,2)-Raney numbers modulo a prime p. For every prime p, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in 0≤ n<p^m. Interestingly, these results fall into three cases: p=2, p=3, and p≥5, and the behaviors in these three cases are very different. For p=2, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For p=3, we characterize exactly the nonzero terms and determine their residues. For p≥ 5, the zero runs are governed by a multiscale system of residue intervals modulo powers of p, from which both the record values and the complete zero-run spectrum are obtained.
ER  -

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