Bergeron’s Fibonacci sign-multiplicity conjecture

Prove that the ungraded sign-character multiplicity of the (1,3)-bosonic-fermionic coinvariant ring R_n^{(1,3)} equals \(\frac12F_{3n}\) for every n\geq1, where F_n is defined by F_0=0, F_1=1, and F_n=F_{n-1}+F_{n-2}.

Background

Bergeron conjectured a closed formula for the dimension of the sign-character component of R_n{(1,3)}. The proposed value is half of the Fibonacci number with index 3n.

The paper proves a separate binomial-sum identity showing that the specialization of the proposed multigraded conjecture equals 12F3n\frac12F_{3n}. However, it does not prove that this value is the actual sign-character multiplicity for all n; the conjecture is used together with the specialized Theta conjecture to explain how equality in the conditional upper bound would follow.

References

While collecting computational evidence, Bergeron conjectured the following.

— The sign character of the triagonal fermionic coinvariant ring  (2501.09920 - Lentfer, 17 Jan 2025) in Conjecture (Conjecture 7.2), Section 7 (One set of bosons and three sets of fermions)

While collecting computational evidence, Bergeron conjectured the following. For all $n \geq 1$, \langle \Frob(R_n{(1,3)};1;1,1,1), s_{(1n)}\rangle = \frac{1}{2}F_{3n}.

— The sign character of the triagonal fermionic coinvariant ring  (2501.09920 - Lentfer, 17 Jan 2025) in Conjecture \ref{conj:bergeron_fibonacci}, Section 1, “One set of bosons and three sets of fermions”