Real-rootedness of reduced multiset Schett polynomials

Prove that every nonzero reduced multiset Schett polynomial \(\widehat S_{M,i}(t)=\sum_{T\in\mathcal{T}_M,\,\lfloor\mathsf{ee}(T)/2\rfloor=i} t^{\lfloor\mathsf{oe}(T)/2\rfloor}\), where \(\mathsf{ee}(T)\) and \(\mathsf{oe}(T)\) count nodes of even outdegree at even and odd levels, respectively, has only real zeros, and develop a tractable insertion operator for these statistics.

Background

The paper identifies a related conjecture concerning reduced multiset Schett polynomials introduced by Lin and Ma. For a weakly increasing tree $T$, the statistics $\mathsf{ee}(T)$ and $\mathsf{oe}(T)$ record nodes of even outdegree at even and odd levels, respectively. The conjectured polynomials refine the tree enumeration by fixing $\lfloor\mathsf{ee}(T)/2\rfloor=i$ and using $\lfloor\mathsf{oe}(T)/2\rfloor$ as the exponent.

The authors explain that their maximal-label deletion method for Eulerian–Narayana polynomials does not directly apply because it does not preserve degree parity at each base vertex or level parity in the attached forests. They therefore identify finding a tractable insertion operator for these statistics as a natural extension, leaving the stated real-zero conjecture unresolved.