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AI-assisted proof of the Elizalde–Luo pattern count

For every n≥1, the number of nonnesting permutations of {1,1,…,n,n} avoiding both 1132 and 3312, under the published containment convention, equals 3ⁿ−3·2ⁿ⁻¹+1.

Sources for this record

What was already known

Elizalde and Luo conjectured this formula after checking cases through n=8; no general proof was recorded in their 2025 paper.

AI involvement AI-centered

Demonstrandum credits Claude agents with proposing the retained proofs and Erlbacher with directing the campaign and auditing its verification gates.

The release credits AI agents with proposing retained constructions or proofs and Erlbacher with directing the campaign; AI-centered is the supported individual-result level.

Importance Notable

Proves a published general counting conjecture in permutation patterns.

Limitations

Matching the formal pattern definitions to the published convention remains a separate human comparison. The earlier archived proof predates the reported Lean formalization.

Technical verification Formal (Lean 4 / mathlib kernel)

The full general-n theorem is reported accepted by Lean 4 with no sorry, admit or native_decide and only standard mathlib axioms. Source-to-Lean definition faithfulness remains a separate human step.

Formal checking

The pinned repository revision reports a Lean 4 kernel proof of the general counting theorem.

Checker
Lean 4/mathlib kernel on theorem elizalde_luo_1132_3312; reported axiom audit
Scope
The general-n Elizalde–Luo {1132,3312} counting formula
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

Human direction and gate auditing are documented, but no named person explains the central reasoning of this individual AI result. Few is the evidence default, not a finding of inability to understand it.

Correction and publication history