Characteristic Chow Polynomials of Layered-Permutation Intervals
Abstract
We study characteristic Chow polynomials of principal intervals in generalized subword order, equivalently of pattern-containment intervals below layered permutations. A terminal-chain recurrence and canonical rightmost embeddings yield a recursive rational generating function for every fixed number of layers, governed by a universal denominator. In the two-layer case we obtain an explicit kernel, prove canonical interlacing when one layer has size at most four, and prove eventual interlacing at every fixed width by a Laguerre limit. We also treat two families with unboundedly many layers. Singleton-prefix intervals have a closed Delannoy-Jacobi gamma polynomial and form real-rooted interlacing sequences. Singleton-sandwich intervals admit rational three-variable series and a marked-slot formula for all gamma coefficients. Their central-layer-two and central-layer-three slices are real-rooted, with interlacing under growth from two to three. For each fixed pair of singleton flanks, a Laguerre differential transform proves eventual interlacing as the central layer grows. These results exhibit composition intervals as a structured testing ground for positivity and root-location questions for Chow polynomials of posets.
Related
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- Sparse Lifts of q-Linearized Permutation Polynomials with Explicit Inverses
BibTeX
@misc{zhang2026characteristic,
title = {{Characteristic Chow Polynomials of Layered-Permutation Intervals}},
author = {Jiahao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33037},
eprint = {2610.33037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33037},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Characteristic Chow Polynomials of Layered-Permutation Intervals
AU - Jiahao Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.33037
UR - https://arxiv.org/abs/2610.33037
AB - We study characteristic Chow polynomials of principal intervals in generalized subword order, equivalently of pattern-containment intervals below layered permutations. A terminal-chain recurrence and canonical rightmost embeddings yield a recursive rational generating function for every fixed number of layers, governed by a universal denominator. In the two-layer case we obtain an explicit kernel, prove canonical interlacing when one layer has size at most four, and prove eventual interlacing at every fixed width by a Laguerre limit. We also treat two families with unboundedly many layers. Singleton-prefix intervals have a closed Delannoy-Jacobi gamma polynomial and form real-rooted interlacing sequences. Singleton-sandwich intervals admit rational three-variable series and a marked-slot formula for all gamma coefficients. Their central-layer-two and central-layer-three slices are real-rooted, with interlacing under growth from two to three. For each fixed pair of singleton flanks, a Laguerre differential transform proves eventual interlacing as the central layer grows. These results exhibit composition intervals as a structured testing ground for positivity and root-location questions for Chow polynomials of posets.
ER -