Counting the markings of a graph that realize racks and quandles
Abstract
A marking of a graph Γ with vertex set V is a map R: V → AutΓ; it makes V into a right quasigroup, and by a theorem of Ta that right quasigroup is a rack exactly when R_(Rᵥ(w))=RᵥR_wRᵥ⁻¹ for all v,w ∈ V. Let μ_(rack)(Γ) and μ_(qnd)(Γ) be the number of markings realizing a rack, respectively a quandle. Ta tabulated these numbers for complete graphs, star graphs and cycles on few vertices and left five entries of the table undetermined. We determine four of them: μ(K₅)=(1708,404),qquad μ(K₆)=(36538,6658), μ(K_(1,5))=(7628,1708),qquad μ(K_(1,6))=(223378,36538), where μ=(μ_(rack),μ_(qnd)). The blind sweep over all |AutΓ|^(|V|) markings is 10²⁰ maps in the largest of these cases; we replace it by a search that propagates the forcing R_(Rᵥ(w))=RᵥR_wRᵥ⁻¹ and whose tree has at most 2.7 · 10⁵ nodes. We also record an identity that is visible in Ta's own printed data but is not remarked on there, μ_(qnd)(K_(1,n-1))=μ_(rack)(Kₙ₋₁), and explain it: the row of the centre of a star is a shift, and racks on a set are in bijection with pairs consisting of a quandle and a compatible shift. The identity relates two of the previously undetermined entries to two others, so the four new values check each other. All statements below are machine-checked in Lean 4.
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Archived files
- Version 2 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
e592feea465de77900d06fb92faf3f14171804b29e7af99d8f6d905526f768ec
Claim ledger
Stated results
rackmark-01known2026-08-23
Definitions 2.1/2.5/3.3 and Theorem 5.1 transcribed; the two forms of the condition proved equivalent over Equiv.Perm; Aut(K_(1,n-1)) fixes the centre for n >= 3
rackmark-02routine2026-08-23
muᵣack and mu_qnd are cardinalities, and a propagating search with proved completeness computes them
rackmark-03known data2026-08-23
all fifteen (Table 1 has 15 printed cells, 5 per row – count corrected 2026-08-28) printed cells of Table 1 reproduced from the source's own definitions
rackmark-04candidate2026-08-23
mu(K₅) = (1708, 404) – the first '?' cell of the Kₙ row
rackmark-05candidate2026-08-23
mu(K₆) = (36538, 6658)
rackmark-06candidate2026-08-23
mu(K_(1,5)) = (7628, 1708) – the first '?' cell of the star row
rackmark-07candidate2026-08-23
mu(K_(1,6)) = (223378, 36538) – the last '?' cell of the star row
rackmark-08candidate2026-08-23
racks are quandles with a shift: the kink bijection, and the star reduction that turns it into mu_qnd(K_(1,n-1)) = muᵣack(Kₙ₋₁)
rackmark-09routine2026-08-23
the conjugation convention is pinned only at n = 4: the opposite reading gives the source's (13,5) at n = 3 and (112,34) against the printed (114,36) at n = 4
rackmark-10routine2026-08-23
the blind (n!)ⁿ sweep – the source's own method – reproduces the search's counts at n <= 4
rackmark-11routine2026-08-23
negative controls: a marking that is not a rack marking, a rack marking that is not a q-marking, a permutation that is a K₄ automorphism but not a star automorphism, and refutations either side of muᵣack(K₅), muᵣack(K₆), muᵣack(K_(1,5)), muᵣack(K_(1,6))
rackmark-12routine2026-08-30
a marking realizes a rack iff it is equivariant for the group its rows generate, over an arbitrary vertex set — with the two corollaries that turn muᵣack into an orbit count: the row at v centralizes the stabilizer of v, and a rack is determined by one row per orbit
rackmark-13routine2026-08-30
controls for the equivariance classification: the closure form is strictly stronger than the rack axiom, and the stabilizer-centralizer corollary is not vacuous
rackmark-14candidate2026-08-30
the cycle row past the source's Table 1: mu(C₈) = (316,16), mu(C₉) = (340,14), mu(C₁0) = (748,19), each quandle coordinate checked against the source's own Proposition 5.9
rackmark-15candidate2026-08-30
cycleRackFormula, a closed form for muᵣack(Cₙ), equal to the honest cardinality at n = 3,...,10 – the instances of the general theorem of rackmark-19
rackmark-16routine2026-08-30
negative controls for the cycle column: refutations either side of muᵣack(C₈) and muᵣack(C₉), and three separations showing the closed form is not a re-dressing of Proposition 5.9
rackmark-17routine2026-08-30
the –load-dynlib route for this family: an import-free mirror of the whole marking search, proved DEFINITIONALLY equal to the original
rackmark-18candidate2026-08-30
mu(K₇) = (1164056, 152900) – the last? cell of Table 1, and with it the whole Kₙ row n = 0..7 in Lean
rackmark-19prose2026-08-30
the closed form muᵣack(Cₙ) = sum over e | n of [J_(n/e)(e) + S(e, n/e)] for every n >= 3, completing the source's Proposition 5.9 on its rack half
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Source. Lực Ta, *Graph quandles: Generalized Cayley graphs of racks and right quasigroups*, arXiv:2506.04437 v5 (2026-03-30), published in the *Journal of Non-Associative Structures* 1 (2026), issue 1 (2026-03-31), jonas:17215. Read from the live v5 HTML (arxiv.org/html/2506.04437v5) on 2026-08-22; the? cells are open in the published version.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7