An inclusion coupling of prefix laws and the Chan-Pak rank inequality for all finite posets
Abstract
Let P be a finite poset, let x∈ P, and let Nk count its linear extensions in which x occupies position k. We prove Chan and Pak's conjecture that, for all integers a,h≥1, (a+h−1)Na+h−1Na ≥ (a−1)Na−1Na+h+hNaNa+h. The key result is an inclusion coupling of the laws of the set of elements preceding x, conditional on consecutive positions of x. We construct reversible exchange chains on the corresponding layers of order ideals and couple their generators so that the two prefix sets remain nested. Fishburn's correlation inequality gives monotonicity of endpoint probabilities; their unit total masses provide the rate bounds needed for the coupling. A stationary distribution of the coupled chain then has the required marginals. Normalized cover observables yield a four-term nonnegative certificate for the case h=1, and the general case follows by telescoping. The argument applies at every width and gives a proof of Stanley's log-concavity inequality with Fishburn's inequality as its sole external inequality.
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BibTeX
@misc{virtanen2026inclusion,
title = {{An inclusion coupling of prefix laws and the Chan-Pak rank inequality for all finite posets}},
author = {Jukka Tapio Virtanen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25227},
eprint = {2610.25227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25227},
}
RIS
TY - UNPB
TI - An inclusion coupling of prefix laws and the Chan-Pak rank inequality for all finite posets
AU - Jukka Tapio Virtanen
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25227
UR - https://arxiv.org/abs/2610.25227
AB - Let P be a finite poset, let x∈ P, and let N_k count its linear extensions in which x occupies position k. We prove Chan and Pak's conjecture that, for all integers a,h≥1, (a+h−1)N_a+h−1N_a ≥ (a−1)N_a−1N_a+h+hN_aN_a+h. The key result is an inclusion coupling of the laws of the set of elements preceding x, conditional on consecutive positions of x. We construct reversible exchange chains on the corresponding layers of order ideals and couple their generators so that the two prefix sets remain nested. Fishburn's correlation inequality gives monotonicity of endpoint probabilities; their unit total masses provide the rate bounds needed for the coupling. A stationary distribution of the coupled chain then has the required marginals. Normalized cover observables yield a four-term nonnegative certificate for the case h=1, and the general case follows by telescoping. The argument applies at every width and gives a proof of Stanley's log-concavity inequality with Fishburn's inequality as its sole external inequality.
ER -