Related papers: Shellable simplicial complex and switching rook po…

Let $\mathcal{P}$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper we give a combinatorial interpretation of the $h$-polynomial of $K[\mathcal{P}]$, showing that it is the rook polynomial of…

Let P be a simple thin polyomino, roughly speaking a polyomino that has no holes and does not contain a square tetromino as a subpolyomino. In this paper, we determine the reduced Hilbert series $h(t)/(1-t)^d$ of K[P] by proving that $h(t)$…

We explore the novel connection between rook placements on collections of cells, also known as pruned chessboards, and the algebraic properties of ideals generated by $2$-minors. We design an algorithm to compute the switching rook…

The rook polynomial is a generating function that enumerates the number of ways to place rooks, with no two in the same row or column, on a collection of cells regarded as a pruned chessboard. In combinatorial commutative algebra, special…

In this article we investigate the shellability of the flag simplicial complexes attached to non-simple and thin polyominoes. As a consequence, we obtain the Cohen-Macaulayness and a combinatorial interepetation of the $h$-polynomial of the…

Grid polyominoes form a class of thin polyominoes with one or more holes arranged in a grid-like pattern in the plane. In this paper, we prove that the rook polynomial of grid polyominoes coincides with the h-polynomial of their…

Let $X$ be a convex polyomino such that its vertex set is a sublattice of $\mathbb{N}^2$. Let $\Bbbk[X]$ be the toric ring (over a field $\Bbbk$) associated to $X$ in the sense of Qureshi, \emph{J. Algebra}, 2012. Write the Hilbert series…

We study the properties of the rook complex $\mathcal{R}$ of a polyomino $\mathcal{P}$ seen as independence complex of a graph $G$, and the associated Stanley--Reisner ideal $I_\mathcal{R}$. In particular, we characterize the polyominoes…

We prove that the Bruhat-Chevalley-Renner order on the rook monoid is EL-shellable. We determine the homeomorphism type of the associated order complex.

Shellings of simplicial complexes have long been a useful tool in topological and algebraic combinatorics. Shellings of a complex expose a large amount of information in a helpful way, but are not easy to construct, often requiring deep…

In geometric, algebraic, and topological combinatorics, the unimodality of combinatorial generating polynomials is frequently studied. Unimodality follows when the polynomial is (real) stable, a property often deduced via the theory of…

We investigate the shellability of the polyhedral join $\mathcal{Z}^*_M (K, L)$ of simplicial complexes $K, M$ and a subcomplex $L \subset K$. We give sufficient conditions and necessary conditions on $(K, L)$ for $\mathcal{Z}^*_M (K, L)$…

We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…

Two are the objectives of the present paper. First we study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and…

Reduction trees are a way of encoding a substitution procedure dictated by the relations of an algebra. We use reduction trees in the subdivision algebra to construct canonical triangulations of flow polytopes which are shellable. We…

We present a conjecture about the reduced Hilbert series of the coordinate ring of a simple polyomino in terms of particular arrangements of non-attacking rooks that can be placed on the polyomino. By using a computational approach, we…

Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular…

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…

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several…

Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial complex K_P which carries complete information about the combinatorial type of P. In the case when P is simple, K_P is the same as dP*, where P*…