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Algebraic combinatorics · Jack polynomials · map and matching enumeration

Matchings–Jack Conjecture

Collaboration beta

After writing the Jack parameter as α = 1 + b, do all Jack connection coefficients have nonnegative integer coefficients in b, as predicted by a combinatorial interpretation using matchings or maps?

Cμ,νλ(b):=cμ,νλ(1+b)∈ℕ[b](λ,μ,ν⊢n)
Known results and sources
Three circular matching diagrams overlap above an incomplete row of amber coefficient tiles.
The conjecture asks whether the integral polynomials indexed by three partitions have only nonnegative coefficients, with a matching or map interpretation.

Research problem

Exact mathematical statement

Put α=1+b\alpha=1+b. For every three partitions λ,μ,ν⊢n\lambda,\mu,\nu\vdash n, let

Cμ,νλ(b):=cμ,νλ(1+b).C_{\mu,\nu}^{\lambda}(b):=c_{\mu,\nu}^{\lambda}(1+b).

The Matchings–Jack conjecture asks whether

Cμ,νλ(b)∈ℕ[b](λ,μ,ν⊢n).C_{\mu,\nu}^{\lambda}(b)\in\mathbb N[b] \qquad(\lambda,\mu,\nu\vdash n).

Published polynomiality and integrality are inputs. The remaining assertion is coefficientwise nonnegativity, together with the desired matching/map interpretation; this workspace does not treat the reported research route as a proof.

Problem infographic

Problem at a glance

Three perfect matchings combine into a ribbon-like map and an open tray of polynomial coefficient cells.
Three partitions index the connection coefficient; the open question is whether every polynomial coefficient is nonnegative and admits the desired combinatorial interpretation.

Current mathematical picture

Where work on Matchings–Jack Conjecture stands

Partially resolved

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureNaive global interleaving and arbitrary raising paths

The Revision-15 interleaved operator has mixed-sign output at b=0, and the source warns that without raising confluence the result can depend on the path from the unit profile. Aligned localized Boolean fibers on a common decorated module, followed by source-authorized target-isolated raising with verified diamonds, remain viable.

Route status · Narrowed route
Main reductionConditional completion theorem

The current work states a ten-condition source-groupoid package under which lexicographic induction gives nonnegative touch-all coefficients and the defect recursion supplies all higher modes.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeRealize the atomic Gate-1 creator crossing with exact source chronology and weights.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Matchings–Jack Conjecture in numbers

5.5kretained lines of mathematical investigation5,520 in the current working snapshot
Argument development
4,573 · 83%
Explored or eliminated routes
183 · 3%
Computational analysis
124 · 2%
Open obligations
275 · 5%
Definitions and setup
365 · 7%
7selected mapped statements1routes 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.

Proposed byIan P. Goulden and David M. JacksonProposed1996Status checkedAug 14, 2026
Source recordedAug 14, 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

13 selected steps

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

13 selected steps

Scroll horizontally to explore the route

Working route overview for Matchings–Jack ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Every Matchings–Jack connection polynomial should be coefficientwise nonnegative — Depends on missing premiseEvery Matchings–Jackconnection polynomial shouldbe…Conditional completion theorem — Depends on missing premiseConditional completiontheoremCurrent reduction — Depends on missing premiseCurrent reductionTwo positive assembly routes — Depends on missing premiseTwo positive assembly routesTwo-generator defect recursion — Depends on missing premiseTwo-generator defectrecursionAligned contextual Boolean factorization — Depends on missing premiseAligned contextual BooleanfactorizationClosing target — Depends on missing premiseClosing targetNaive global interleaving and arbitrary raising paths — stoppedNaive global interleavingand arbitrary raising pathsRealize the atomic Gate-1 creator crossing with exact source chronology and weights. — OpenRealize the atomic Gate-1creator crossing with exactsource…Construct positive target-isolated Gate-0 raising and prove its first labelled diamond. — OpenConstruct positivetarget-isolated Gate-0raising…Prove point/unpoint, cross-gate, quotient and completion compatibility for one operator. — OpenProve point/unpoint,cross-gate, quotient andcompletion…Positive Gate-0 target isolation — OpenPositive Gate-0 targetisolationRaising diamond condition — OpenRaising diamond condition
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.

Explored alternatives

Other routes

1 recorded
Narrowed routeNaive global interleaving and arbitrary raising paths

The Revision-15 interleaved operator has mixed-sign output at b=0, and the source warns that without raising confluence the result can depend on the path from the unit profile. Aligned localized Boolean fibers on a common decorated module, followed by source-authorized target-isolated raising with verified diamonds, remain viable.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

5 featured tasks
01
Realize the atomic Gate-1 creator crossing with exact source chronology and weights.Suggested move: Use the asymmetric fixture in work order MJ-R16-A to prove or refute the physical 1+b,1,b crossing while preserving directed boundary words, roots, labels and deletion order.
Ready to work on
02
Construct positive target-isolated Gate-0 raising and prove its first labelled diamond.Suggested move: Derive the full homogeneous sign ledger for the (1)→(2), (1,1)→(2,1) fixtures and compare both labelled paths to (2,2) before taking any quotient.
Ready to work on
03
Positive Gate-0 target isolation

A source correspondence must isolate each raised coefficient with a nonnegative right-hand side built only from already constructed data; the algebraic flow alone does not do this.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Raising diamond condition

Distinct labelled face-raising orders must produce the same weighted object, including lower-degree corrections, or the resulting coefficient may depend on the chosen path.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
05
Prove point/unpoint, cross-gate, quotient and completion compatibility for one operator.Suggested move: Audit the equal-unit-face fixture with a nontrivial automorphism, preserving roots and labels until orbit accounting is complete.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusPartially resolved

The general coefficientwise-nonnegativity and matching-interpretation conjecture remains open. Polynomiality and integrality are known, the top-degree coefficient is positive, and important labelled or partition-specialized cases are proved.

[2][3][4]
External progress

What the literature has established

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

  1. Peer reviewedLabelled and one-partition-equal-to-(n) variants and cases were proved, while the general problem was still described as open.[4]
  2. PreprintBen Dali proved the integrality part, upgrading known rational polynomiality to integer coefficients.[3]
  3. Peer reviewedThe top-degree coefficient was proved to be a positive integer; the general conjecture remained open.[2]
  4. Historical sourceGoulden and Jackson introduced the coefficients and the nonnegative-integral matching interpretation conjecture.[1]
4 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusMatchings–Jack conjecture
Solved special caseTop-degree Matchings–Jack coefficient

The coefficient of highest possible degree is a positive integer.

[2]
Weaker or relaxed formIntegrality part of the conjecture

All connection coefficients lie in Z[b], but integrality alone does not imply coefficientwise nonnegativity.

[3]
Solved special caseLabelled matching variants and special cases

Important labelled and one-partition-equal-to-(n) cases have a positive combinatorial interpretation.

[4]

Formalization opportunities

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

  • Formalization targetA formal library treatment of Jack symmetric functions and the exact connection coefficients used here was not identified.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

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.

5 standing statements2 proposed statements5 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction4 of 74
  • lemma2 of 72
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementEvery Matchings–Jack connection polynomial should be coefficientwise nonnegative
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementTwo-generator defect recursionintermediate
  • retained route statementAligned contextual Boolean factorizationintermediate
  • retained route statementTwo positive assembly routesintermediate
  • retained route statementConditional completion theoremintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureNaive global interleaving and arbitrary raising pathsreported failure
  • Research targetRealize the atomic Gate-1 creator crossing with exact source chronology and weights.open
  • Research targetConstruct positive target-isolated Gate-0 raising and prove its first labelled diamond.open
  • Research targetProve point/unpoint, cross-gate, quotient and completion compatibility for one operator.open
  • Research targetPositive Gate-0 target isolationopen
  • Research targetRaising diamond conditionopen
  • Narrowed routeNaive global interleaving and arbitrary raising pathsThe Revision-15 interleaved operator has mixed-sign output at b=0, and the source warns that without raising confluence the result can depend on the path from the unit profile. Aligned localized Boolean fibers on a common decorated module, followed by source-authorized target-isolated raising with verified diamonds, remain viable.
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 bridgeRealize the atomic Gate-1 creator crossing with exact source chronology and weights.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

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.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

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Prepared starting pointRealize the atomic Gate-1 creator crossing with exact source chronology and weights.

Matchings–Jack Conjecture · ready to start

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Research contextPrepared context for any AI agent

After writing the Jack parameter as α = 1 + b, do all Jack connection coefficients have nonnegative integer coefficients in b, as predicted by a combinatorial interpretation using matchings or maps?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Connection Coefficients, Matchings, Maps and Combinatorial Conjectures for Jack Symmetric Functionsoriginal source · Ian P. Goulden, David M. Jackson · Transactions of the American Mathematical Society · 1996 · accessed Aug 14, 2026
  2. 2
    The Top-Degree Part in the Matchings-Jack Conjecturepeer reviewed result · Adam Burchardt · The Electronic Journal of Combinatorics · 2021-05-07 · DOI 10.37236/9191 · accessed Aug 14, 2026
  3. 3
    Integrality in the Matching-Jack conjecture and the Farahat-Higman algebrapreprint · Houcine Ben Dali · arXiv · 2022-03-28 · ARXIV 2203.14879 · accessed Aug 14, 2026
  4. 4
    On the Matchings-Jack and Hypermap-Jack Conjectures for Labelled Matchings and Star Hypermapspeer reviewed result · Andrei L. Kanunnikov, Valentin V. Promyslov, Ekaterina A. Vassilieva · The Electronic Journal of Combinatorics · 2024-07-26 · DOI 10.37236/9575 · accessed Aug 14, 2026

Important qualifications

  • Scoped to primary author-hosted, journal-hosted, and arXiv records; no exhaustive citation search was attempted.
  • External metadata is independent context and does not review source-reported reductions.
  • No formalization or maintained computation was identified in the scoped search.

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