Let In be the set of involutions in the symmetric group Sn, and for A⊆{0,1,...,n}, letFnA={σ∈In|σ has a fixed points for some a∈A}. We give a complete characterisation of the sets A for which FnA, ...with the order induced by the Bruhat order on Sn, is a graded poset. In particular, we prove that Fn{1} (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When FnA is graded, we give its rank function. We also give a short new proof of the EL-shellability of Fn{0} (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.
The authors compute the（rational） Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes ...p.
Monomial Ideals of Forest Type Jahan, Ali Soleyman; Zheng, Xinxian
Communications in algebra,
08/2012, Volume:
40, Issue:
8
Journal Article
Peer reviewed
As a generalization of the facet ideal of a forest, we define monomial ideal of forest type and show that monomial ideals of forest type are pretty clean. As a consequence, we show that if I is a ...monomial ideal of forest type in the polynomial ring S, then Stanley's decomposition conjecture holds for S/I. The other main result of this article shows that a clutter is totally balanced if and only if it has the free vertex property, and which is also equivalent to say that its edge ideal is a monomial ideal of forest type or is generated by an M sequence.
Spheres arising from multicomplexes Murai, Satoshi
Journal of combinatorial theory. Series A,
11/2011, Volume:
118, Issue:
8
Journal Article
Peer reviewed
Open access
In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex Δ on the vertex set
V with
Δ
≠
2
V
, the ...deleted join of Δ with its Alexander dual
Δ
∨
is a combinatorial sphere. In this paper, we extend Bierʼs construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory.
Let
I
n
be the set of involutions in the symmetric group
S
n
, and for
A
⊆
{
0
,
1
,
…
,
n
}
, let
F
n
A
=
{
σ
∈
I
n
∣
σ
has
exactly
a
fixed
points
for
some
a
∈
A
}
.
We give a complete ...characterisation of the sets
A
for which
F
n
A
, with the order induced by the Bruhat order on
S
n
, is a graded poset. In particular, we prove that
F
n
{
1
}
(i.e. the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When
F
n
A
is graded, we give its rank function. We also give a short, new proof of the EL-shellability of
F
n
{
0
}
(i.e. the set of fixed-point-free involutions), recently proved by Can et al.
First we prove that certain complexes on directed acyclic graphs are shellable. Then we study independence complexes. Two theorems used for breaking and gluing such complexes are proved and applied ...to generalize the results by Kozlov.
An interesting special case is anti-Rips complexes: a subset
P
of a metric space is the vertex set of the complex, and we include as a simplex each subset of
P
with no pair of points within distance
r
. For any finite subset
P
of
R
the homotopy type of the anti-Rips complex is determined.
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define
shellability of chain ...complexes
in a similar way as for cell complexes and introduce the notion of
regular
chain complexes. In the case of chain complexes coming from simplicial complexes we recover the classical notions but, in contrast to the topological case, in the abstract setting shellings turn out to be a weaker homological invariant. In particular, we study special chain complexes, which are
cones
, and a class of regular chain complexes, which we call
totally regular
, for which we can obtain complete homological information if an additional condition is fulfilled.
In this article we introduce the m-cover poset of an arbitrary bounded poset P, which is a certain subposet of the m-fold direct product of P with itself. Its ground set consists of multichains of P ...that contain at most three different elements, one of which has to be the least element of P, and the other two elements have to form a cover relation in P. We study the m-cover poset from a structural and topological point of view. In particular, we characterize the posets whose m-cover poset is a lattice for all m>0, and we characterize the special cases, where these lattices are EL-shellable, left-modular, or trim. Subsequently, we investigate the m-cover poset of the Tamari lattice Tn, and we show that the smallest lattice that contains the m-cover poset of Tn is isomorphic to the m-Tamari lattice Tn(m) introduced by Bergeron and Préville-Ratelle. We conclude this article with a conjectural desription of an explicit realization of Tn(m) in terms of m-tuples of Dyck paths.
12 pages, 3 figures
Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for ...some $a \in A$}\}. \ We give a complete characterisation of the sets $A$ for which $F_n^A$, with the order induced by the Bruhat order on $S_n$, is a graded poset. In particular, we prove that $F_n^{\{1\}}$ (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When $F_n^A$ is graded, we give its rank function. We also give a short new proof of the EL-shellability of $F_n^{\{0\}}$ (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.
Soit $I_n$ l’ensemble d’involutions dans le groupe symétrique $S_n$, et pour $A \subseteq \{0,1,\ldots,n\}$, soit\ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ a $a$ points fixes pour quelque $a \in A$}\}. \ Nous caractérisons tous les ensembles $A$ dont les $F_n^A$ , avec l’ordre induit par l’ordre de Bruhat sur $S_n$, est un posetgradué. En particulier, nous démontrons que $F_n^{\{1\}}$ (c’est-à-dire, l’ensemble d’involutions avec précis en point fixe)est gradué, ce qui résout une conjecture d’Hultman à l’affirmative. Lorsque $F_n^A$ est gradué, nous donnons sa fonctionde rang. En plus, nous donnons une nouvelle démonstration courte l’EL-shellability de $F_n^{\{0\}}$ (c’est-à-dire, l’ensembled’involutions sans points fixes), établie récemment par Can, Cherniavsky et Twelbeck.
A polytopal digraph G(P) is an orientation of the skeleton of a convex polytope P. The possible non-degenerate pivot operations of the simplex method in solving a linear program over P can be ...represented as a special polytopal digraph known as an LP digraph. Presently there is no general characterization of which polytopal digraphs are LP digraphs, although four necessary properties are known: acyclicity, unique sink orientation (USO), the Holt–Klee property and the shelling property. The shelling property was introduced by Avis and Moriyama (2009), where two examples are given in d=4 dimensions of polytopal digraphs satisfying the first three properties but not the shelling property. The smaller of these examples has n=7 vertices. Avis, Miyata and Moriyama (2009) constructed for each d⩾4 and n⩾d+2, a d-polytope P with n vertices which has a polytopal digraph which is an acyclic USO that satisfies the Holt–Klee property, but does not satisfy the shelling property. The construction was based on a minimal such example, which has d=4 and n=6. In this paper we explore the shelling condition further. First we give an apparently stronger definition of the shelling property, which we then prove is equivalent to the original definition. Using this stronger condition we are able to give a more general construction of such families. In particular, we show that given any 4-dimensional polytope P with n0 vertices whose unique sink is simple, we can extend P for any d⩾4 and n⩾n0+d−4 to a d-polytope with these properties that has n vertices. Finally we investigate the strength of the shelling condition for d-crosspolytopes, for which Develin (2004) has given a complete characterization of LP orientations.
PDF
