Cycle Counting and Character Expectations Using Alternating Structures

Abstract

Recently, two related papers [arXiv:2412.13941, arXiv:2409.03626] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [arXiv:1410.2540] using stackings and by [arXiv:1410.2579] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [arXiv:2412.13941] and [arXiv:2409.03626]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two.

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BibTeX RIS

BibTeX

@misc{shma2026cycle,
  title = {{Cycle Counting and Character Expectations Using Alternating Structures}},
  author = {Noam Ta Shma},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27593},
  eprint = {2608.27593},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27593},
}

RIS

TY  - UNPB
TI  - Cycle Counting and Character Expectations Using Alternating Structures
AU  - Noam Ta Shma
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.27593
UR  - https://arxiv.org/abs/2608.27593
AB  - Recently, two related papers [arXiv:2412.13941, arXiv:2409.03626] found a connection between two subjects: the w-cycle theorem, which is a theorem about counting appearances of cycles reading out a word w in certain graphs, and character expectations on word measures. The w-cycle theorem was proven independently by [arXiv:1410.2540] using stackings and by [arXiv:1410.2579] using bislim structures. In the current work, we generalize stackings and bislim structures to alternating stackings and alternating bislim structures. We show how this significantly strengthens the w-cycle theorem for words admitting such alternating structures, and as a result, also strengthens the recent results of [arXiv:2412.13941] and [arXiv:2409.03626]. We show that generic words admit alternating bislim structures, and therefore, the strengthened results hold for generic words. Using our new machinery, we address conjectures of Wilton, of Hanany-Puder and of Puder-Shomroni. We prove that all three conjectures hold for generic words, but we also find counterexamples for the first two.
ER  -

Abstract

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