Real-rootedness and interlacing for parking functions and Chow polynomials
Abstract
We prove real-rootedness for the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying an interlacing-preserving transition of M.~Leander. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric g-contributions of Ehrenborg--Hetyei--Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q.~Xiao also proved it by a different differential recurrence. We also give a second proof of the individual statement, by finite Schur--Szegő convolution, that does not use the common interlacer. These results prove Conjecture~4.2 of Xiao and Conjecture~11.2 of Ehrenborg--Hetyei--Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.
Related
- Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
- The real-rootedness of the toric g-contribution polynomials
- Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
- Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart h*-polynomials
- A total variation threshold for the first coordinates of uniform parking functions via Lorentzian polynomials
BibTeX
@misc{alexandersson2026realrootedness,
title = {{Real-rootedness and interlacing for parking functions and Chow polynomials}},
author = {Per Alexandersson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05131},
eprint = {2609.05131},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05131},
}
RIS
TY - UNPB
TI - Real-rootedness and interlacing for parking functions and Chow polynomials
AU - Per Alexandersson
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.05131
UR - https://arxiv.org/abs/2609.05131
AB - We prove real-rootedness for the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying an interlacing-preserving transition of M.~Leander. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric g-contributions of Ehrenborg--Hetyei--Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q.~Xiao also proved it by a different differential recurrence. We also give a second proof of the individual statement, by finite Schur--Szegő convolution, that does not use the common interlacer. These results prove Conjecture~4.2 of Xiao and Conjecture~11.2 of Ehrenborg--Hetyei--Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.
ER -