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

Jaideep Sai Padhi

For every fixed r,s≥1, both restricted-permutation count sequences a_(r,s)(n), forbidding π_(i+r)−π_i=s, and b_(r,s)(n), forbidding the corresponding absolute difference, are claimed to be holonomic. The argument uses an interleaving bijection with finitely many seams and D-finite closure.

Sources for this record

What was already known

Spahn and Zeilberger posed general holonomicity for r,s>1. Several small cases were known, including a guessed a_(2,2) recurrence.

AI involvement AI-led

Claude supplied the interleaving and holonomicity argument. Padhi selected the problem, tested cases and rejected model errors; no complete independent human proof check is documented.

Claude supplied the interleaving and holonomicity argument under Padhi’s direction.

Importance Notable

Reports an all-parameter answer to the stated restricted-permutation holonomicity challenge.

Limitations

Brute-force checks cover seven (r,s) pairs for n=4–8 and seam indices for nine pairs; they do not verify the general bijection or D-finite closure. The separate guessed a_(2,2) recurrence has only a partial proof in this paper.

Technical verification Partial

The paper checks the bijection by brute-force enumeration for seven (r,s) pairs and n=4–8 and tests seam indices for nine pairs. These finite checks do not verify all r,s or the D-finite closure argument; no completed full check is documented.

Human understanding Few

No independent human account of the model-generated all-parameter bijection and D-finite closure is documented; Few is the evidence default.

Correction and publication history