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arXiv · 2411.04123

The monoid representation of upho posets and total positivity

Ziyao Fu·Yulin Peng·Yuchong Zhang
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Abstract

We show that all totally positive formal power series with integer coefficients and constant term $1$ are precisely the rank-generating functions of Schur-positive upho posets, thereby resolving the main conjecture proposed by Gao, Guo, Seetharaman, and Seidel. To achieve this, we construct a bijection between finitary colored upho posets and atomic, left-cancellative, invertible-free monoids, which restricts to a correspondence between $\mathbb{N}$-graded colored upho posets and left-cancellative homogeneous monoids. Furthermore, we introduce semi-upho posets and develop a convolution operation on colored upho posets with colored semi-upho posets within this monoid-theoretic framework.

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Ziyao Fu, Yulin Peng, Yuchong Zhang. 2024-11-06. The monoid representation of upho posets and total positivity. https://arxiv.org/abs/2411.04123

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