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Combinatorics · Latin squares · algebraic enumeration

Alon–Tarsi Latin-Square Conjecture

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Do even and odd Latin squares fail to cancel in every even order?

Δn≠0for every evenn
Known results and sources
A text-free mathematical cover representing Alon–Tarsi Latin-Square Conjecture.
Do even and odd Latin squares fail to cancel in every even order?

Research problem

Exact mathematical statement

For an even positive integer nn, let Δn\Delta_n be the sum over all labeled Latin squares of order nn of the product of all row and column permutation signs. The Alon–Tarsi Latin-Square Conjecture asks whether

Δn≠0for every evenn.\Delta_n\ne0\qquad\text{for every even }n.

The current source keeps this target open. It organizes four source-reported routes—trade and sparse-anchor capacity, Boolean determinant cores, squared-Plücker adjacent order, and cyclic symmetry—but supplies none of their universal closing interfaces.

Problem infographic

Problem at a glance

A text-free scientific explainer showing the objects, constraints, and unresolved routes for Alon–Tarsi Latin-Square Conjecture.
Four source-reported routes end at explicit open interfaces: adjacent-order noncancellation, sparse-anchor capacity, cyclic free-orbit depth, and an adaptive determinant-core residue.

Current mathematical picture

Where work on Alon–Tarsi Latin-Square Conjecture stands

Partially resolved

The cumulative v4 source keeps the Alon–Tarsi target open, separates four exact source-reported routes, records thirty-three retained interfaces, and replaces three disproved overbroad endpoints with explicit adjacent-order, sparse-anchor, cyclic, and adaptive-core obligations.

Leading routeAdjacent-order cycle packets

Prioritize the exact scalar adjacent-order interface, derive the first surviving cycle-packet quotient, and seek straightening or prime separation.

Route status · Active route
Useful failureExperimental three-view odd-cycle proposal

The compiled restart explicitly supersedes that experimental proposal and makes the fixed-row residual the first exact reduction. Use the exact fixed-row residual with union Fourier trades, cross-switches, and reserved positive pair-flip anchors.

Route status · Eliminated route
Main reductionFour exact source-reported routes reduce the target to explicit open interfaces

The v4 source organizes trade/capacity, Boolean-core, adjacent-order, and cyclic-descent routes, each ending at an explicit open noncancellation or capacity interface.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeAdjacent-order noncancellation

Prove that (2r-2)! Theta_r differs from AT*(2r-1) for every r at least 2.

Task status · Ready to work on
Later mathematical updatev4 records four route interfaces and retires three overbroad endpoints

The cumulative v4 source keeps the target open, records four explicit route interfaces, and retires sharp-valuation, parity-weight, and all-fixed-anchor endpoints.

v4 source revision order; not occurrence time or public priority

Work mapped so far

Alon–Tarsi Latin-Square Conjecture in numbers

2.2kretained lines of mathematical investigation1,241 in the current working snapshot
Argument development
1,846 · 83%
Explored or eliminated routes
46 · 2%
Computational analysis
104 · 5%
Open obligations
66 · 3%
Definitions and setup
175 · 8%
40selected mapped statements4routes investigated5open questions5contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedAug 28, 2026
Source recordedAug 30, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

22 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

22 selected steps

Scroll horizontally to explore the route

Working route overview for Alon–Tarsi Latin-Square ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Even-order signed Latin-square difference noncancellation target — Depends on missing premiseEven-order signedLatin-square differencenoncancellation…Four exact source-reported routes reduce the target to explicit open interfaces — Depends on missing premiseFour exact source-reportedroutes reduce the target toexplicit…CYCLIC-DESCENT-1 — Depends on missing premiseCYCLIC-DESCENT-1CYCLIC-ORBIT-1 — Depends on missing premiseCYCLIC-ORBIT-1CYCLIC-TRACE-1 — Depends on missing premiseCYCLIC-TRACE-1DET-ADJACENT-1 — Depends on missing premiseDET-ADJACENT-1DET-CORE-1 — Depends on missing premiseDET-CORE-1DET-RANKFACT-1 — Depends on missing premiseDET-RANKFACT-1The v4 snapshot has four exact source-reported routes — Depends on missing premiseThe v4 snapshot has fourexact source-reported routesTR-CROSS-1 — Depends on missing premiseTR-CROSS-1TR-SPARSE-ANCHOR-6 — Depends on missing premiseTR-SPARSE-ANCHOR-6TR-SPARSE-ANCHOR-LEMMA — Depends on missing premiseTR-SPARSE-ANCHOR-LEMMAAdjacent-order cycle packets — activeAdjacent-order cycle packetsSparse-anchor capacity — activeSparse-anchor capacityRetired sharp determinant congruence — stoppedRetired sharp determinantcongruenceRetired parity-only weighted Plücker identification — stoppedRetired parity-only weightedPlücker identificationRetired all-fixed-anchor Hall endpoint — stoppedRetired all-fixed-anchorHall endpointAdjacent-order noncancellation — OpenAdjacent-ordernoncancellationUniversal sparse-anchor capacity — OpenUniversal sparse-anchorcapacityCyclic free-orbit valuation — OpenCyclic free-orbit valuationUniversal trade move coverage — OpenUniversal trade movecoverageAdaptive determinant-core residue — OpenAdaptive determinant-coreresidue
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeAdjacent-order cycle packets

Prioritize the exact scalar adjacent-order interface, derive the first surviving cycle-packet quotient, and seek straightening or prime separation.

Route status · Active route
Active routeSparse-anchor capacity

Characterize sparse anchors intrinsically, express load by move blocks, and prove overload peeling terminates while retaining strict surplus.

Route status · Active route

Explored alternatives

Other routes

2 recorded
Eliminated routeExperimental three-view odd-cycle proposal

The compiled restart explicitly supersedes that experimental proposal and makes the fixed-row residual the first exact reduction. Use the exact fixed-row residual with union Fourier trades, cross-switches, and reserved positive pair-flip anchors.

Route status · Eliminated route
Eliminated routeUnreserved Hall matching

The source states that unreserved Hall can prove only nonnegativity; positive fixed anchors must be reserved. Keep ordinary Hall as a cancellation component, but pair it with an explicit strict-surplus argument.

Route status · Eliminated route

Route statements and reductions

Statements the next route can inspect and build on

Route statementTR-SPARSE-ANCHOR-LEMMA

Local capacity inequalities imply reserved Hall and strict surplus, as source-reported.

Source-reported route statement · dependencies incomplete
Route statementDET-ADJACENT-1

Exact adjacent-order factorization, as source-reported.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Adjacent-order noncancellation

Prove that (2r-2)! Theta_r differs from AT*(2r-1) for every r at least 2.

Suggested move: Prove that (2r-2)! Theta_r differs from AT*(2r-1) for every r at least 2.
Ready to work on
02
Universal sparse-anchor capacity

Construct a nonempty sparse positive anchor set satisfying every nonanchor capacity inequality in all even orders.

Suggested move: Construct a nonempty sparse positive anchor set satisfying every nonanchor capacity inequality in all even orders.
Ready to work on
03
Cyclic free-orbit valuation

Prove the stated p-adic depth for the cyclic free-orbit term and supply a descent/coverage scheme reaching every even order.

Suggested move: Prove the stated p-adic depth for the cyclic free-orbit term and supply a descent/coverage scheme reaching every even order.
Ready to work on
04
Universal trade move coverage

Show every negative residual marked square has an admissible sign-reversing union or cross move; this remains necessary but not sufficient for capacity.

Suggested move: Show every negative residual marked square has an admissible sign-reversing union or cross move; this remains necessary but not sufficient for capacity.
Ready to work on
05
Adaptive determinant-core residue

Produce a nonzero order-dependent residue or inequality after the false universal sharp valuation has been retired.

Suggested move: Produce a nonzero order-dependent residue or inequality after the false universal sharp valuation has been retired.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 28, 2026
Current statusPartially resolved

The cited paper proves the signed nonvanishing assertion for the infinite family of even orders 2^r p with p prime. The all-even-order statement remains open in this bounded review.

[1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedDrisko proved the conjecture for even orders of the form 2^r p with p prime, a proper family of even orders rather than all even orders.[1]
1 cited sources0 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA formalization needs parity conventions for Latin squares and an exact finite signed enumeration interface for every even order.
  • Formalization targetThe known 2^r p family must remain a solved special case and not be generalized automatically to all even orders.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

v4 records four route interfaces and retires three overbroad endpointsThe cumulative v4 source keeps the target open, records four explicit route interfaces, and retires sharp-valuation, parity-weight, and all-fixed-anchor endpoints.

Changed the research frontierLater mathematical revision

v4 source revision order; not occurrence time or public priority

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

38 standing statements2 proposed statements5 open questions
Statements by mathematical role40 selected mapped statements
  • equivalence1 of 401
  • lemma37 of 4037
  • theorem candidate1 of 401
  • reduction1 of 401
Selected mathematical clusters1 mathematical clusters
Current research mapThe exact target remains open. The v4 source records four explicit routes, thirty-three source-reported retained interfaces, four controlling blockers, and three material corrections that retire overbroad endpoints without invalidating their narrower foundations.53 displayed rows · 2 routes included
  • retained route statementAT-HYBRID-1
  • retained route statementAT-NORM-1
  • retained route statementAT-PARITY-1
  • retained route statementAT-RED-1
  • retained route statementAny one of four explicit source-reported interfaces would close the targetintermediate
  • retained route statementEven-order signed Latin-square difference noncancellation target
  • retained route statementCYCLIC-DESCENT-1
  • retained route statementCYCLIC-DIV-1/2
  • retained route statementCYCLIC-ORBIT-1
  • retained route statementCYCLIC-TRACE-1
  • retained route statementDET-2CLASS-1
  • retained route statementDET-ADJACENT-1
  • retained route statementDET-BOOL-1
  • retained route statementDET-BORDER-1
  • retained route statementDET-COMP-1
  • retained route statementDET-CORANK-1
  • retained route statementDET-CORE-1
  • retained route statementDET-DEFECT-1
  • retained route statementDET-FIXSLICE-1
  • retained route statementDET-ODD-DIV-1
  • retained route statementDET-PATHHAF-1
  • retained route statementDET-PERM-1
  • retained route statementDET-PLUCKER-1
  • retained route statementDET-PRINCIPAL-1
  • retained route statementDET-R2HAF-1
  • retained route statementDET-RANKFACT-1
  • retained route statementFixed-row residualintermediate
  • retained route statementThe v4 snapshot has four exact source-reported routesintermediate
  • retained route statementPositive anchor familyintermediate
  • retained route statementFour exact source-reported routes reduce the target to explicit open interfacesintermediate
  • retained route statementSigned targetintermediate
  • retained route statementTR-ANCHOR-1
  • retained route statementTR-CROSS-1
  • retained route statementTR-HALL-6
  • retained route statementTR-MARKED-NORM-1
  • retained route statementTR-ROW-1
  • retained route statementTR-SPARSE-ANCHOR-6
  • retained route statementTR-SPARSE-ANCHOR-LEMMA
  • retained route statementTR-SPLICE-1
  • retained route statementTR-UNION-1/2
  • DerivationThe governing source states that any one of its four explicit remaining interfaces would close the exact signed-difference target; this remains a proposed source-reported route, not an accepted proof.proposed
  • Recorded relationshipThe v4 source reports four routes toward the exact target and explicitly leaves each route's final interface open.supports · reported by source
  • Recorded relationshipThe current four-route inventory supports the new four-route reduction while the retired three-route inventory remains historical.supports · reported by source
  • Useful failureRetired all-fixed-anchor Hall endpointreported failure
  • Useful failureRetired parity-only weighted Plücker identificationreported failure
  • Useful failureRetired sharp determinant congruencereported failure
  • Research targetAdjacent-order noncancellationopen
  • Research targetUniversal sparse-anchor capacityopen
  • Research targetUniversal trade move coverageopen
  • Research targetAdaptive determinant-core residueopen
  • Research targetCyclic free-orbit valuationopen
  • Active routeAdjacent-order cycle packetsPrioritize the exact scalar adjacent-order interface, derive the first surviving cycle-packet quotient, and seek straightening or prime separation.
  • Active routeSparse-anchor capacityCharacterize sparse anchors intrinsically, express load by move blocks, and prove overload peeling terminates while retaining strict surplus.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve that (2r-2)! Theta_r differs from AT*(2r-1) for every r at least 2.

The current research map records this as an open mathematical step.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Provide the exact source-stated universal interface without relying on a retired endpoint.
  • Preserve the open target and source-reported evidence posture until separately reviewed.

Continue the mathematics

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Prepared starting pointAdjacent-order noncancellation

Alon–Tarsi Latin-Square Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Do even and odd Latin squares fail to cancel in every even order?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references1 cited works · next context review by Nov 28, 2026

The mathematical context was checked on Aug 28, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Proof of the Alon-Tarsi Conjecture for n=2^r ppeer reviewed result · Arthur A. Drisko · The Electronic Journal of Combinatorics · 1998-05-10 · DOI 10.37236/1366 · accessed Aug 28, 2026

Important qualifications

  • This was a bounded primary-source and publisher-record search, not an exhaustive literature, priority, citation, rights, or authorship review.
  • Open status means that the cited source states or studies the problem as a conjecture or open problem and the bounded search found no statement-aligned primary resolution; it does not prove that no later claim exists.
  • Recent preprints are recorded only with their stated preprint posture and are not treated as peer-reviewed or independently verified.
  • No submitted attachment, submitted URL, packet-reported computation, or model output was treated as independent external authority.
  • No statement-aligned formalization, certificate, or independently reproduced computation was established by this search.

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