Pith Number

pith:YZV3SH3Y

pith:2026:YZV3SH3YZDTL2L3DIQJMOEQOVR

not attested not anchored not stored refs resolved

Rooted bicubic planar maps via Dyck paths

Jackie N. Kaminski, Juan B. Gil

Rooted bicubic planar maps on 2n vertices correspond bijectively to Dyck paths of semilength 3n with colored ascents of length divisible by 3.

arxiv:2605.17515 v1 · 2026-05-17 · math.CO

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\pithnumber{YZV3SH3YZDTL2L3DIQJMOEQOVR}

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Record completeness

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4 Citations open

5 Replications open

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Claims

C1strongest claim

We establish an explicit bijection between rooted bicubic planar maps on 2n vertices and Dyck paths of semilength 3n with ascents of length divisible by 3, where each 3j-ascent is colored using one of g_j colors corresponding to the rooted 3-connected bicubic maps on 2j vertices.

C2weakest assumption

The construction assumes that the numbers g_j of rooted 3-connected bicubic maps on 2j vertices are already known or recursively available independently of the full map enumeration, so that the coloring step does not presuppose the decomposition being proved.

C3one line summary

An explicit bijection maps rooted bicubic planar maps on 2n vertices to colored Dyck paths of semilength 3n, proving Tutte's decomposition into 3-connected components via Bell transformations.

References

8 extracted · 8 resolved · 0 Pith anchors

[1] J.-L. Baril, R. Genestier, A. Giorgetti, and A. Petrossian, Rooted planar maps modulo some patterns, Discrete Math.339(2016), 1199–1205 2016

[2] D. Birmajer, J. Gil, P. McNamara, and M. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials,Lattice Path Combinatorics and Applications, G.E. Andrews, C. Krattenthaler, A. Krinik ( 2019

[3] Batagelj, An inductive definition of the class of 3-connected quadrangulations of the plane,Discrete Math.78(1989), 45–53 1989

[4] D. Birmajer, J. Gil, and M. Weiner, A family of Bell transformations,Discrete Math.342(2019), 38–54 2019

[5] E. Horev, M. J. Katz, R. Krakovski, A. Nakamoto, Polychromatic 4-coloring of cubic bipartite plane graphs,Discrete Math.312(2012), 715–719. ROOTED BICUBIC PLANAR MAPS VIA DYCK PATHS 17 Figure 27.Roote 2012

Receipt and verification

First computed	2026-05-20T00:04:43.230488Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c66bb91f78c8e6bd2f634412c7120eac4f849938301f11e5a2123d0f8cfe9a29

Aliases

arxiv: 2605.17515 · arxiv_version: 2605.17515v1 · doi: 10.48550/arxiv.2605.17515 · pith_short_12: YZV3SH3YZDTL · pith_short_16: YZV3SH3YZDTL2L3D · pith_short_8: YZV3SH3Y

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/YZV3SH3YZDTL2L3DIQJMOEQOVR \
  | 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: c66bb91f78c8e6bd2f634412c7120eac4f849938301f11e5a2123d0f8cfe9a29

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f5bf455d7247aa89007f591adf8779724d2f0cb8472c24a8abeb55324b8ad032",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CO",
    "submitted_at": "2026-05-17T15:59:46Z",
    "title_canon_sha256": "ec18d003ed15f1c9b7c807a7880858ac2df5bab07e7472dd46795d61bc2a0f23"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.17515",
    "kind": "arxiv",
    "version": 1
  }
}