Enumeration of ballot-admissible Fib(n,k) ribbons

Enumerate ballot-admissible Fib(n,k) ribbons for every integer n 4, with the goal of extending the enumeration established for ballot-admissible Fib(3,k) ribbons.

Background

The paper constructs a bijection between ballot-admissible Fib(3,k) Fibonacci ribbon tableaux and peakless Motzkin paths, thereby proving their equinumerosity. It then identifies the broader enumeration of ballot-admissible Fib(n,k) ribbons for n 4 as unresolved. The authors suggest that the techniques developed in the paper may help establish a recursive enumeration in this general case.

References

However, the problem of enumerating ballot-admissible $\text{Fib}(n,k)$ ribbons for $n\ge 4$ remains an open problem.

— A bijection between peakless Motzkin paths and LR tableaux  (2608.13401 - Tenn, 13 Aug 2026) in Section Conclusion and Future Work