Relaxed coparking functions and Stanley's conjecture for matroid h-vectors

@misc{oh2026relaxed,
  title = {{Relaxed coparking functions and Stanley's conjecture for matroid $h$-vectors}},
  author = {SuHo Oh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17446},
  eprint = {2609.17446},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17446},
}

TY  - UNPB
TI  - Relaxed coparking functions and Stanley's conjecture for matroid h-vectors
AU  - SuHo Oh
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.17446
UR  - https://arxiv.org/abs/2609.17446
AB  - Stanley's conjecture asserts that the h-vector of a matroid is a pure O-sequence. Corry, Dochtermann, McClain, Perkinson and Yi introduced cycle systems, which give a bijective proof for the matroids that admit one, through coparking functions, and proposed generalized cycle systems. Every matroid admitting a cycle system is regular, and the Wagner and Petersen graphs admit no generalized cycle system consisting of circuits. We propose a relaxation. Where the coparking recursion breaks down, at the strata whose unique union is independent, the coparking functions are replaced by a fibre: a pure multicomplex with the h-vector of the dead node, the minor of the matroid attached to the stratum. We prove, for any matroid with a fixed basis, that whenever the required fibres exist the relaxed coparking functions form a pure multicomplex whose degree sequence is the h-vector of the matroid, so Stanley's conjecture follows. The proof rests on a version of Dhar's burning algorithm for a matroid with a fixed basis, which gives the purity, and on a deletion-contraction identity that computes the gap between the h-vector and the coparking functions as a sum of local h-vectors, one per dead node. Coned, biconed and triconed graphs carry a fibre system with their canonical spanning trees, as do the Wagner and Petersen graphs. Beyond graphs, every basis of a matroid of corank two, of a matroid of rank at most four or of a uniform matroid carries a fibre system, so Stanley's conjecture follows for those classes. We also give a graph of radius two on twelve vertices which, with its breadth-first spanning tree, carries no fibre system.
ER  -