Solutions to Five Challenge Problems in Enumerative and Algorithmic Combinatorics, with an Account of the Human-Machine Methodology Employed

Jaideep Sai Padhi

For three-row tableaux with runs of length at least two, G(n)~C₁8ⁿ/n⁴; for odd runs, H(n)~C₂(7+5√2)ⁿ/n⁴. The constants are positive finite boundary sums, with numerical estimates about 0.52128605909 and 0.6389278129.

Sources for this record

What was already known

Kauers and Zeilberger conjectured both growth rates and gave coarse numerical estimates. The proposed proof imports cone-walk estimates and adapts them to Markov-modulated walks through a corrector ladder.

AI involvement AI-led

Claude developed the local-limit and asymptotic argument. Padhi selected the challenge, ran computations, applied controls and corrected model errors.

Claude generated the local-limit and asymptotic argument under Padhi’s problem selection and validation.

Importance Notable

Reports solutions of Kauers–Zeilberger Conjectures 2a and 2b.

Limitations

Exact enumeration and symbolic checks cover selected initial terms and intermediate identities. The transported local-limit argument lacks a documented complete proof check, and the displayed decimal constants are extrapolations without certified error bounds.

Technical verification Partial

Exact enumeration reproduces published initial terms and independent extrapolations support constants; symbolic algebra checks some covariance and corrector identities. The transported local-limit theorem and full asymptotic proof have no documented completed rigorous review.

Independent checking

Source-reported symbolic algebra checks of exact intermediate identities

Checker
Jaideep Sai Padhi, paper author, executing model-written code
Scope
Selected transfer-matrix covariance and odd-run corrector identities in Section 9
Basis
Reported by the linked checker
Evidence source

Human understanding Few

The paper explains the formal argument, but it credits Claude with the mathematics and does not identify a human who independently explains the Markov-additive corrector ladder; Few is the evidence default.

Correction and publication history