scholarly journals Maximal Fillings of Moon Polyominoes, Simplicial Complexes, and Schubert Polynomials

10.37236/1167 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Luis Serrano ◽  
Christian Stump
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Canonical Connection ◽  
Cyclic Sieving Phenomenon ◽  
North East ◽  
Schubert Polynomials ◽  
Dyck Paths ◽  
Vertex Decomposable ◽  
Cyclic Sieving

We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we  show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Moreover, for Ferrers shapes we construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths of length $2(n-2k)$. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to the language of $k$-flagged tableaux with promotion.

scholarly journals Generalized triangulations, pipe dreams, and simplicial spheres

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Luis Serrano ◽  
Christian Stump
Keyword(s):  
Simplicial Complex ◽  
Canonical Connection ◽  
Cyclic Sieving Phenomenon ◽  
North East ◽  
Schubert Polynomials ◽  
Dyck Paths ◽  
Nous Montrons ◽  
International Audience ◽  
Vertex Decomposable ◽  
Cyclic Sieving

International audience We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable and thus a shellable sphere. In particular, this implies a positivity result for Schubert polynomials. For Ferrers shapes, we moreover construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to $k$-flagged tableaux with promotion. Nous décrivons un lien canonique entre les $(0,1)$-remplissages maximaux d'un polyomino-lune évitant les chaînes Nord-Est d'une longueur donnée, et les "pipe dreams'' réduits d'une certaine permutation. En suivant cette approche nous montrons que le complexe simplicial de tels remplissages maximaux est une sphère "vertex-decomposable'' et donc "shellable''. En particulier, cela entraîne un résultat de positivité sur les polynômes de Schubert. De plus, nous construisons, dans le cas des diagrammes de Ferrers, une bijection vers les remplissages maximaux évitant les chaînes Sud-Est de même longueur, qui se spécialise en une bijection entre les $k$-triangulations d'un $n$-gone et les $k$-faisceaux de chemins de Dyck. A l'aide de celle-ci, nous traduisons une instance conjecturale du phénomène de tamis cyclique pour les $k$-triangulations avec rotation dans le cadre des tableaux $k$-marqués avec promotion.

scholarly journals Simplicial Complexes of Whisker Type

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia
Keyword(s):  
Simplicial Complex ◽  
Short Proof ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Linear Resolution ◽  
Cw Complex ◽  
Alexander Dual ◽  
Vertex Decomposable ◽  
The Ideal

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

scholarly journals Balanced Vertex Decomposable Simplicial Complexes and their h-vectors

10.37236/2552 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jennifer Biermann ◽  
Adam Van Tuyl
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Finite Simplicial Complex ◽  
Vertex Decomposable ◽  
Flag Complexes ◽  
Special Case

Given any finite simplicial complex $\Delta$, we show how to construct from a colouring $\chi$ of $\Delta$ a new simplicial complex $\Delta_{\chi}$ that is balanced and vertex decomposable. In addition, the $h$-vector of $\Delta_{\chi}$ is precisely the $f$-vector of $\Delta$.  Our construction generalizes the "whiskering'' construction of Villarreal, and Cook and Nagel. We also reverse this construction to prove a special case of a conjecture of Cook and Nagel, and Constantinescu and Varbaro on the $h$-vectors of flag complexes.

scholarly journals The Cyclic Sieving Phenomenon on Circular Dyck Paths

10.37236/8720 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Per Alexandersson ◽  
Svante Linusson ◽  
Samu Potka
Keyword(s):  
Cyclic Group ◽  
Catalan Numbers ◽  
Natural Action ◽  
Cyclic Sieving Phenomenon ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
Cyclic Sieving

We give a $q$-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this $q$-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Möbius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.

scholarly journals A note on the van der Waerden complex

2019 ◽  
Vol 124 (2) ◽  
pp. 179-187
Author(s):  
Becky Hooper ◽  
Adam Van Tuyl
Keyword(s):  
Simplicial Complex ◽  
Commutative Algebra ◽  
Simplicial Complexes ◽  
Arithmetic Progressions ◽  
Combinatorial Commutative Algebra ◽  
Vertex Decomposable

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

scholarly journals Optimal Decision Trees on Simplicial Complexes

10.37236/1900 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Decision Tree ◽  
Decision Trees ◽  
Simplicial Complex ◽  
Elementary Theory ◽  
Simplicial Complexes ◽  
Optimal Decision ◽  
Property A ◽  
Recursive Definition ◽  
Topological Combinatorics ◽  
Definition Of

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension $i$ with respect to a decision tree on a simplicial complex is greater than or equal to the $i$th reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an "optimal" decision tree such that equality holds for each $i$; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.

scholarly journals The Cyclic Sieving Phenomenon for Non-Crossing Forests

10.37236/2419 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Stefan Kluge
Keyword(s):  
Cyclic Group ◽  
Cyclic Sieving Phenomenon ◽  
Cyclic Sieving

In this paper we prove that the set of non-crossing forests together with a cyclic group acting on it by rotation and a natural q-analogue of the formula for their number exhibits the cyclic sieving phenomenon, as conjectured by Alan Guo.

Spectral gap of a weighted 3-simplicial complex

2021 ◽  
pp. 2150084
Author(s):  
Khalid Hatim ◽  
Azeddine Baalal
Keyword(s):  
Lower Bound ◽  
Simplicial Complex ◽  
Upper Bound ◽  
Spectral Gap ◽  
Simplicial Complexes ◽  
Cheeger Constant ◽  
Nonzero Eigenvalue ◽  
New Framework

In this paper, we construct a new framework that’s we call the weighted [Formula: see text]-simplicial complex and we define its spectral gap. An upper bound for our spectral gap is given by generalizing the Cheeger constant. The lower bound for our spectral gap is obtained from the first nonzero eigenvalue of the Laplacian acting on the functions of certain weighted [Formula: see text]-simplicial complexes.

Subdivisions of Simplicial Complexes Preserving the Metric Topology

2012 ◽  
Vol 55 (1) ◽  
pp. 157-163 ◽  
Author(s):  
Kotaro Mine ◽  
Katsuro Sakai
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Simplicial Subdivision ◽  
Metric Topology ◽  
The One

AbstractLet |K| be the metric polyhedron of a simplicial complex K. In this paper, we characterize a simplicial subdivision K′ of K preserving the metric topology for |K| as the one such that the set K′(0) of vertices of K′ is discrete in |K|. We also prove that two such subdivisions of K have such a common subdivision.

Sign in / Sign up

Export Citation Format

Share Document