Pith Number

pith:OOWYULWN

pith:2026:OOWYULWNNHTGIXBG6EBI72722E

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Burnside process on parking functions and Dyck paths

Ivan Z. Feng, J. E. Paguyo

The Burnside process on parking functions and labeled Dyck paths mixes in O(n log n) steps and yields uniform sampling algorithms.

arxiv:2605.16244 v1 · 2026-05-15 · math.PR · math.CO

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

Our main result shows that both processes are rapidly mixing, with mixing times upper bounded by O(n log n).

C2weakest assumption

The specific group actions (S_n permuting coordinates for parking functions and labels for Dyck paths) allow the general Burnside process to have uniform stationary distribution on orbits and to mix rapidly under the stated dynamics, as described in the abstract for the two special cases.

C3one line summary

Burnside processes on parking functions and Dyck paths mix in O(n log n) time, yielding approximate uniform sampling algorithms for increasing parking functions, Dyck paths, and polygon triangulations.

References

47 extracted · 47 resolved · 2 Pith anchors

[1] Aldous,Mixing times for the branch-rotation chain on cladograms (or the triangulation walk), https://www.stat.berkeley.edu/~aldous/Research/OP/clad-mix.pdf, (2003) 2003

[2] D. Aldous and J. Fill,Reversible Markov chains and random walks on graphs, https://www.stat.berkeley.edu/~aldous/RWG/book.pdf, (2002) 2002

[3] Faster Mixing for Triangulations via Transport Flows 2026 · arXiv:2605.02067

[4] Exact uniform sampling over catalan structures 2018 · arXiv:1803.03945

[5] H. C. Andersen and P. Diaconis,Hit and run as a unifying device, Journal de la Soci´ et´ e Francaise Statistique, 148 (4) (2007), 5-28 2007

Receipt and verification

First computed	2026-05-20T00:01:59.808464Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

73ad8a2ecd69e6645c26f1028febfad105ef50663b2075330c186c8686f557b6

Aliases

arxiv: 2605.16244 · arxiv_version: 2605.16244v1 · doi: 10.48550/arxiv.2605.16244 · pith_short_12: OOWYULWNNHTG · pith_short_16: OOWYULWNNHTGIXBG · pith_short_8: OOWYULWN

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OOWYULWNNHTGIXBG6EBI72722E \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 73ad8a2ecd69e6645c26f1028febfad105ef50663b2075330c186c8686f557b6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3a68423b3039916e47f8706e197b01c0832daa73d4ee6a33fa3896b7e2b971e1",
    "cross_cats_sorted": [
      "math.CO"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-15T17:49:11Z",
    "title_canon_sha256": "8ab681cc07bd2a3c39798632792eb5beebf1ac2866b0f9a9cb4cb337d9da069a"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16244",
    "kind": "arxiv",
    "version": 1
  }
}