The monoid representation of upho posets and total positivity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916495827664896 |
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| author | Fu, Ziyao Peng, Yulin Zhang, Yuchong |
| author_facet | Fu, Ziyao Peng, Yulin Zhang, Yuchong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04123 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The monoid representation of upho posets and total positivity Fu, Ziyao Peng, Yulin Zhang, Yuchong Combinatorics 05A15, 05E05, 06A11, 06A07 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. |
| title | The monoid representation of upho posets and total positivity |
| topic | Combinatorics 05A15, 05E05, 06A11, 06A07 |
| url | https://arxiv.org/abs/2411.04123 |