Prove the collection of explicit conjectural evaluations, due to Cigler and to Cigler–Krattenthaler, for shifted Hankel determinants of Catalan-like sequences arising from weighted enumerations of nonintersecting Motzkin meanders. The determinants have the general form det(a_i+j+s)_0≤ i,j<n, but the supplied sources do not state the individual sequences, parameter ranges, or conjectured right-hand sides precisely enough to record the separate identities.

Cigler and Cigler–Krattenthaler Hankel determinant conjectures

The problem concerns a collection of explicit Hankel-determinant conjectures associated with Johann Cigler and Cigler–Krattenthaler. Earlier work established special cases and left several general identities conjectural.

Known results

Markus Fulmek, 2024: a bijective preprint proof establishes Cigler’s conjecture for Hankel determinants of convoluted Catalan numbers, including vanishing ranges and shifted identities.
Johann Cigler, 2024: an independent preprint proof derives the same Catalan-convolution results and analogous Narayana-polynomial identities.
Earlier computations verified the relevant conjectures through bounded parameter ranges, including $r \le 31$ , but did not prove the general statements.

August 27, 2026 preprint

A newly reported preprint, “Hankel determinants of Catalan-like sequences,” claims determinant evaluations that prove earlier conjectures and confirm an additional binomial determinant identity. This would settle multiple members of the collection, but the claims remain unverified and unrefereed.

Current status (as of August 2026): Several component conjectures have preprint proofs, and a newer preprint claims to settle multiple further identities, but the overall collection is not independently verified.

Cigler and Cigler–Krattenthaler Hankel determinant conjectures

Primary source

Additional references

Progress summary

Solutions 0