Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

In this paper, we introduce polytopes $${\mathscr {B}}_G$$ B G arising from root systems $$B_n$$ B n and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that $${\mathscr {B}}_G$$ B G is reflexive if and only if G is bipartite. Moreover, in the case, $${\mathscr {B}}_G$$ B G has a regular unimodular triangulation. This implies that the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is palindromic and unimodal when G is bipartite. Furthermore, we discuss stronger properties, namely the $$\gamma $$ γ -positivity and the real-rootedness of the $$h^*$$ h ∗ -polynomials. In fact, if G is bipartite, then the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is $$\gamma $$ γ -positive and its $$\gamma $$ γ -polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The $$h^*$$ h ∗ -polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers–Stanley conjecture, we construct a bipartite graph G whose $$h^*$$ h ∗ -polynomial is not real-rooted but $$\gamma $$ γ -positive, and coincides with the h-polynomial of a flag triangulation of a sphere.

$k$-shellable simplicial complexes and graphs

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying an expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture.Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

On Stanley Depths of Certain Monomial Factor Algebras

Let S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

Values and bounds for the Stanley depth

We give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables.

Stanley depth of monomial ideals with small number of generators

For a monomial ideal I ⊂ S = K[x 1...,x n], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x 1,x 2,x 3] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x 1,x 2,x 3]//I) +1 for any monomial ideal in I ⊂ K[x 1,x 2,x 3].