Real-rooted Eulerian polynomials from permutations, words, and paths

RESUMO

We study six Eulerian-type polynomial families. We prove that the descent polynomials of derangements are real-rooted, settling the derangement part of a conjecture of S.~Fu, Z.~Lin, and J.~Zeng. The proof uses a compatible-pair recursion and finite-symbol stability. We also resolve the real-rootedness conjecture in OEIS \oeis{A335340}, strengthen the known rowwise real-rootedness of an even-top descent family to consecutive strict interlacing, and prove real-rootedness, consecutive interlacing, and real-rooted gamma-polynomials for U.~Shankar's super-Eulerian polynomials. A differential recurrence gives consecutive weak interlacing for ternary words counted by increasing runs. Finally, we prove stability of the peak-value refinement and consecutive interleaving of its positive weighted diagonals, settling a conjecture of P.~Alexandersson and O.~Nabawanda.