We show that Stanley's conjecture holds for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we prove that the monomial ideals with all associated primes of height two, are Stanley ideals.
In this paper, we define and characterize the f-graphs. Also, we give a construction of f-graphs and importantly we show that the f-graphs obtained from this construction are Cohen–Macaulay.
We show that the γ-vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In particular, we prove that such γ-vector is the f -vector of some balanced simplicial complex. Moreover, we show that the local γ-vector of the interval subdivision of a simplex is nonnegative; answering a question by Juhnke-Kubitzke et al.Conjecture 1.1 is a strengthening of the well known Charney-Devis conjecture. The Gal conjecture holds for all Coxeter complexes (see [Ste08]), for the dual simplicial complexes of associahedron and cyclohedron (see [NP11]), and for barycentric subdivision of homology sphere (see[NPT11]). The authors in [NP11] conjectured further strengthening of Gal conjecture. Conjecture 1.2. [NP11, Problem 6.4] If ∆ is a flag homology sphere then γ(∆) is the f -vector of some balanced simplicial complex.This conjecture holds for the dual simplicial complex of all flag nestohedera, see in [Ais14]. Frohmader [Fro08, Theorem 1.1] showed that the f -vector of any flag simplicial complex satisfies the Frankl-Füredi-Kalai (FFK) inequalities (see [FFK88]). In [NPT11], authors showed that the γ-vector of the barycentric subdivision of a homology sphere satisfies the FFK inequalities, i.e., the f -vector of a balanced simplicial complex. The first aim of this paper is the confirmation of Conjecture 1.2 in the case of the interval subdivision of a homology sphere. The main theorem is stated as: Theorem 1.3. If ∆ is a simplicial complex with a nonnegative and symmetric h-vector, then the γ-vector of the interval subdivision of ∆ is the f -vector of a balanced simplicial complex.
In this paper, we introduce the concept of spanning simplicial complexes ∆ s (G) associated to a simple finite connected graph G. We give the characterization of all spanning trees of the uni-cyclic graph U n,m . In particular, we give the formula for computing the Hilbert series and h-vector of the Stanley Riesner ring k ∆ s (U n,m ) . Finally, we prove that the spanning simplicial complex ∆ s (U n,m ) is shifted hence ∆ s (U n,m ) is shellable.
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