分布格J(mn)上链多项式的实根性研究
The Real-Rootedness of Chain Polynomials on Distributive Lattice J(mn)
DOI: 10.12677/AAM.2022.1110749, PDF,   
作者: 杨荣涛：西南大学数学与统计学院，重庆
关键词: 偏序集；h-多项式；链多项式；实根性；Poset； h-Polynomial； Chain Polynomial； Real-Rootedness
摘要: 链多项式是定义在偏序集上的一类重要多项式。本文主要研究分布格J(mn)上的链多项式的实根性，通过限制J(mn)中元素的秩得到一个新的偏序集，并且证明了这个新偏序集的链多项式以及h-多项式是实根的。
Abstract: Chain polynomial is one of the most important polynomials defined on posets. This paper mainly studies the real-rootedness of chain polynomial on distributive lattice J(mn). We obtain a new poset by limiting the rank of elements in J(mn), and then prove that the chain polynomial and h-polynomial of this new poset are real-rooted.
文章引用：杨荣涛. 分布格J(mn)上链多项式的实根性研究[J]. 应用数学进展, 2022, 11(10): 7060-7066. https://doi.org/10.12677/AAM.2022.1110749

参考文献

[1] Stanley, R.P. (1989) Log-Concave and Unimodal Sequences in Algebra, Combinatorics, Andgeometry. Annals of the New York Academy of Sciences, 576, 500-535. [Google Scholar] [CrossRef] 
[2] Brenti, F. (1994) Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry: An Update. Jerusalem Com-binatorics’93. Contemporary Mathematics, 178, 71-89. [Google Scholar] [CrossRef] 
[3] Huh, J. (2012) Milnor Numbers of Projective Hypersurfaces and the Chromatic Polynomial of Graphs. Journal of the American Mathematical Society, 25, 907-927. [Google Scholar] [CrossRef] 
[4] Branden, P. (2015) Unimodality, Log-Concavity, Real-Rootedness and Beyond. Mathematics, 437-483.
[5] Mcconville, T., Sagan, B. and Smyth, C. (2021) On a Rank-Unimodality Conjecture of Morier-Genoud and Ovsienko. Discrete Mathematics, 344, Ar-ticle ID: 112483. [Google Scholar] [CrossRef] 
[6] Oguz, E.K. and Ravichandran, M. (2021) Rank Polynomials of Fence Posets Are Unimodal. Arxiv: 2112.00518vl.
[7] Haglund, J. and Zhang, P.B. (2019) Re-al-Rootedness of Variations of Eulerian Polynomials. Advances in Applied Mathematics, 109, 38-54. [Google Scholar] [CrossRef] 
[8] Zhang, P.B. (2016) On the Real-Rootedness of the Local h-Polynomials of Edgewise Subdivisions. The Electronic Journal of Combinatorics, 26.
[9] Stanley, R.P. (1998) Graph Colorings and Related Symmetric Functions: Ideas and Applications: A Description of Results, Interesting Appli-cations & Notable Open Problems. Discrete Mathematics, 193, 267-286. [Google Scholar] [CrossRef] 
[10] Neggers, J. (1978) Representations of Finite Partially Or-dered Sets. Information Systems, 3, 113-133.
[11] Stembridge, J.R. (2007) Counterexamples to the Poset Conjectures of Neggers, Stanley, and Stembridge. Transactions of the American Mathematical Society, 359, 1115-1128. [Google Scholar] [CrossRef] 
[12] Branden, P. (2004) Counterexamples to the Neggers-Stanley Conjecture. American Mathematical Society, 10, 115-158. [Google Scholar] [CrossRef] 
[13] Stanley, R. (2011) Enumerative Combinatorics (Volume 1, Cambridge Studies in Advanced Mathematics 49). 2nd Edition, Cambridge University Press, Cambridge.
[14] Savage, C.D. and Visontai, M. (2015) The s-Eulerian Polynomials Have Only Real Roots. Transactions of the American Math-ematical Society, 367, 1441-1446. [Google Scholar] [CrossRef] 

为你推荐