Matroid flat counts are not unimodal

Alexander Divoux; Chayim Lowen; Shouda Wang

Matroid flat counts are not unimodal

Combinatorics 2026-07-24 v1

Authors: Alexander Divoux , Chayim Lowen , Shouda Wang

Abstract

We give counterexamples to Rota's 1970 conjecture that the sequence counting flats of varying rank in a matroid is unimodal. More specifically, inspired by Larson's recent disproof of the stronger log-concavity conjecture of Mason, we explain a mechanism which turns failures of log-concavity for flats into failures of unimodality under suitable conditions.

Cite

@article{arxiv.2607.22515,
  title  = {Matroid flat counts are not unimodal},
  author = {Alexander Divoux and Chayim Lowen and Shouda Wang},
  journal= {arXiv preprint arXiv:2607.22515},
  year   = {2026}
}

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8 pages