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

Abstract

We study seven 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. We also resolve the real-rootedness conjecture in 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. 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. Finally, we prove that the zig-zag Eulerian polynomials, which enumerate swaps in alternating permutations, have simple negative roots, settling the real-rootedness conjecture of T.~Kyle Petersen and Y.~Zhuang.

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BibTeX RIS

BibTeX

@misc{alexandersson2026realrooted,
  title = {{Real-rooted Eulerian polynomials from permutations, words, and paths}},
  author = {Per Alexandersson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.07325},
  eprint = {2609.07325},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07325},
}

RIS

TY  - UNPB
TI  - Real-rooted Eulerian polynomials from permutations, words, and paths
AU  - Per Alexandersson
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.07325
UR  - https://arxiv.org/abs/2609.07325
AB  - We study seven 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. We also resolve the real-rootedness conjecture in 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. 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. Finally, we prove that the zig-zag Eulerian polynomials, which enumerate swaps in alternating permutations, have simple negative roots, settling the real-rootedness conjecture of T.~Kyle Petersen and Y.~Zhuang.
ER  -

Abstract

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