Multivariate stability of powered triangular recurrences and parametrised Eulerian polynomials

Abstract

We prove multivariate stability for a class of affine triangular recurrences whose two coefficients are raised to an arbitrary positive integer power. The variables keep track of cross-step heights in a weighted lattice-path model. The applications include stability and real rootedness of parametrised Eulerian, Stirling, and Lah families. For the parametrised Eulerian polynomials, we obtain multivariate refinements of known real-rootedness results and prove simplicity and strict interlacing of consecutive rows. A kernel-preserving bijection with increasing binary trees yields a synchronised Foata--Strehl action on pairs and a combinatorial interpretation of their gamma coefficients along with a recurrence. The same bijection identifies refinement posets that admit symmetric Boolean and chain decompositions for parameter one and cover-supported sl₂ operators prove their PECKness. We extend the stability and strict interlacing results to synchronised Stirling permutations and prove PECKness for ordinary Stirling permutations with a fixed number of plateaux.

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BibTeX RIS

BibTeX

@misc{shankar2026multivariate,
  title = {{Multivariate stability of powered triangular recurrences and parametrised Eulerian polynomials}},
  author = {Umesh Shankar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15651},
  eprint = {2609.15651},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15651},
}

RIS

TY  - UNPB
TI  - Multivariate stability of powered triangular recurrences and parametrised Eulerian polynomials
AU  - Umesh Shankar
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15651
UR  - https://arxiv.org/abs/2609.15651
AB  - We prove multivariate stability for a class of affine triangular recurrences whose two coefficients are raised to an arbitrary positive integer power. The variables keep track of cross-step heights in a weighted lattice-path model. The applications include stability and real rootedness of parametrised Eulerian, Stirling, and Lah families. For the parametrised Eulerian polynomials, we obtain multivariate refinements of known real-rootedness results and prove simplicity and strict interlacing of consecutive rows. A kernel-preserving bijection with increasing binary trees yields a synchronised Foata--Strehl action on pairs and a combinatorial interpretation of their gamma coefficients along with a recurrence. The same bijection identifies refinement posets that admit symmetric Boolean and chain decompositions for parameter one and cover-supported sl_2 operators prove their PECKness. We extend the stability and strict interlacing results to synchronised Stirling permutations and prove PECKness for ordinary Stirling permutations with a fixed number of plateaux.
ER  -

Abstract

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