On the Neggers-Stanley Conjecture and the Eulerian Polynomials
On the Neggers-Stanley Conjecture and the Eulerian Polynomials
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DOI：
10.1006/jcta.1997.2839
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发表时间：
1998-05
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影响因子：
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通讯作者：
Vesselin Gasharov
中科院分区：
文献类型：
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作者：
Vesselin Gasharov
本文用组合方法证明了阶为1或2的自然标号分次偏序集(反链的阶数为0)的Thew-多项式是单峰的，从而进一步支持了Neggers-Stanley猜想。对于这样的偏序集，我们还得到了Thew-多项式是对称的组合证明。给出了欧拉多项式是对数凹的单峰多项式的组合证明，并构造了具有模为Δ的斯坦利-赖斯纳理想的外代数模的希尔伯特函数是欧拉数序列的单纯复Δ，从而为Brenti的一个结果提供了一个组合证明.
Abstract We prove combinatorially that theW-polynomials of naturally labeled graded posets of rank 1 or 2 (an antichain has rank 0) are unimodal, thus providing further supporting evidence for the Neggers–Stanley conjecture. For such posets we also obtain a combinatorial proof that theW-polynomials are symmetric. Combinatorial proofs that the Eulerian polynomials are log-concave and unimodal are given and we construct a simplicial complexΔwith the property that the Hilbert function of the exterior algebra modulo the Stanley–Reisner ideal ofΔis the sequence of Eulerian numbers, thus providing a combinatorial proof of a result of Brenti.