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Zeros and interlacing for multiset Eulerian-Narayana polynomials
Abstract
We prove the real-rootedness conjecture of Lin, Ma, Ma, and Zhou (2021) for multiset Eulerian-Narayana polynomials, including simplicity and negativity of all zeros after removing the factor t. Using a forest decomposition and Lagrange inversion, we derive a differential formula and a first-order factorization of the differential operator. We use these factors to show that the nonconstant gamma-coefficients and ordered gamma-zeros are nondecreasing, with corresponding bounds given by the Narayana and Eulerian polynomials. The same factorization gives totally nonnegative coefficient transition matrices. We also prove strict interlacing of the zeros of generating polynomials arising from adjacent columns of the transition matrices.
Related
- Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1
- Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels
- Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
- Real-rootedness and interlacing for parking functions and Chow polynomials
- The (p,q)-Analogues of Bi-Stirling Eulerian Polynomials via continued fractions
BibTeX
@misc{zhang2026zeros,
title = {{Zeros and interlacing for multiset Eulerian-Narayana polynomials}},
author = {Philip B. Zhang and Tongyuan Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00966},
eprint = {2610.00966},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00966},
}
RIS
TY - UNPB
TI - Zeros and interlacing for multiset Eulerian-Narayana polynomials
AU - Philip B. Zhang
AU - Tongyuan Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00966
UR - https://arxiv.org/abs/2610.00966
AB - We prove the real-rootedness conjecture of Lin, Ma, Ma, and Zhou (2021) for multiset Eulerian-Narayana polynomials, including simplicity and negativity of all zeros after removing the factor t. Using a forest decomposition and Lagrange inversion, we derive a differential formula and a first-order factorization of the differential operator. We use these factors to show that the nonconstant gamma-coefficients and ordered gamma-zeros are nondecreasing, with corresponding bounds given by the Narayana and Eulerian polynomials. The same factorization gives totally nonnegative coefficient transition matrices. We also prove strict interlacing of the zeros of generating polynomials arising from adjacent columns of the transition matrices.
ER -