Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and ... more Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by $\Phi_{k+r}(n_1,n_2,...,n_r)$, where $\Phi_{\ell}(n_1,n_2,...,n_r)= \sum_{\substack{1\le{}s_i\le{}n_i s_1+...+s_r=\ell}} \prod_{i=1}^r\binom{n_i}{s_i}$, $\ell\ge{}r$. Fukuda and Weibel have also shown that the upper bound mentioned above is tight for $d\ge{}4$, $2\le{}r\le{}\lfloor\frac{d}{2}\rfloor$, and for all $0\le{}k\le{}\lfloor\frac{d}{2}\rfloor-r$. In this paper we construct a set of $r$ neighborly $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $2\le{}r\le{}d-1$, for which the upper bound of Fukuda and Weibel is attained for all $0\le{}k\le{}\lfloor\frac{d+r-1}{2}\rfloor-r$. Our approach is based on what is known as the Cayley trick for Minkowski sums. A direct consequence of our r...
Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated... more Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Brändén and Solus have given sufficient conditions under which the image of a polynomial under a certain operator associated to barycentric subdivision has such a decomposition. This paper gives a new proof of their result which generalizes to subdivision operators in the setting of uniform triangulations of simplicial complexes, introduced by the first named author. Sufficient conditions under which these decompositions are also interlacing are described. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in f -vector theory arise from this work.
In this paper we present a bijection between two well known families of Catalan objects: the set ... more In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid\left\langle v,\alpha \right\rangle=m\}$ as separating wall. As a result, it restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.
Motivated by the relation N m (C n ) = (mn+1)N m (A n−1 ), holding for the m-generalized Catalan ... more Motivated by the relation N m (C n ) = (mn+1)N m (A n−1 ), holding for the m-generalized Catalan numbers of type A and C, the connection between dominant regions of the m-Shi arrangement of type A n−1 and C n is investigated. More precisely, it is explicitly shown how mn + 1 copies of each element of the set R m + (A n−1 ) of dominant regions of the m-Shi arrangement of type A n−1 , biject onto the set R m + (C n ) of type C n such regions. This is achieved by exploiting two different viewpoints to express the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between mn + 1 copies of each m-Dyck path of height n and the set of N − E lattice paths inside an n × mn rectangle is provided.
We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski su... more We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 + ⋯ + P r , of r convex d-polytopes P 1 , . . . , P r in R d , where d ≥ 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal . In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f -and h-vector calculus and shellings, and generalizes the methodology used in [15] and [14] for proving upper bounds on the f -vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P 1 + ⋯ + P r as a section of the Cayley polytope C of the summands; bounding the k-faces of P 1 + ⋯ + P r reduces to bounding the subset of the (k + r − 1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds. arXiv:1502.02265v1 [cs.CG] 8 Feb 2015 vertices or number of facets of the summands. Fogel, Halperin, and Weibel [6] extended one of the results in [7], and expressed the number of facets of the Minkowski sum of r 3-polytopes in terms of the number of facets of the summands. Quite recently Weibel [21] provided a relation for the number of k-faces of the Minkowski sum of r ≥ d summands in terms of the k-faces of the Minkowski sums of subsets of size d − 1 of these summands. This result should be viewed in conjunction with a result by Sanyal [19] stating that the number of vertices of the Minkowski sum of r d-polytopes, where r ≥ d, is strictly less than the product of the vertices of the summands (whereas for r ≤ d−1 this is indeed possible). About 3 years ago, the authors of this paper proved the first tight upper bound on the number of k-faces for the Minkowski sum of two d-polytopes in R d , for any d ≥ 2 and for all 0 ≤ k ≤ ), a result which was subsequently extended to three summands in collaboration with Konaxis cf. ).
In this paper we construct a set of $r$ neighborly $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}... more In this paper we construct a set of $r$ neighborly $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $2\le{}r\le{}d-1$, for which the upper bound of Fukuda and Weibel is attained for all $0\le{}k\le{}\lfloor\frac{d+r-1}{2}\rfloor-r$. Our approach is based on what is known as the Cayley trick for Minkowski sums. A direct consequence of our result is a tight asymptotic bound on the complexity of the Minkowski sum $P_1+P_2+...+P_r$, for any fixed dimension $d$ and any $2\le{}r\le{}d-1$, when the number of vertices of the polytopes is (asymptotically) the same.
In this paper we present a bijection $\omega_n$ between two well known families of Catalan object... more In this paper we present a bijection $\omega_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, $\omega_n$ bijects the facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid<v,\alpha >=m\}$ as separating wall. As a result, $\omega_n$ restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. The map $\omega_n$ is a composition of two bijections in which integer partitions in an $m$-staircase shape come into play.
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, 2012
We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski su... more We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 ⊕ P 2 , of two d-dimensional convex polytopes P 1 and P 2 , as a function of the number of vertices of the polytopes.
We end with algorithmic considerations, and we show how our tight bounds for the parallel polytop... more We end with algorithmic considerations, and we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in $\mathbb{E}^d$.
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13, 2013
We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski su... more We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 + P 2 + P 3 , of three d-dimensional convex polytopes P 1 , P 2 and P 3 , as a function of the number of vertices of the polytopes, for any d ≥ 2. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope C, the problem of counting the number of k-faces of P 1 + P 2 + P 3 , reduces to counting the number of (k + 2)-faces of the subset of C comprising of the faces that contain at least one vertex from each P i . In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of r d-polytopes, where r ≥ d. For d ≥ 4, the maximum values are attained when P 1 , P 2 and P 3 are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.
Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011
Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a fixed number of m distinct radii... more Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a fixed number of m distinct radii ρ 1 , ρ 2 , . . . , ρ m , we show that the worst-case combinatorial complexity of the convex hull
Let Φ be an finite root system with corresponding reflection group W and let m be a nonnegative i... more Let Φ be an finite root system with corresponding reflection group W and let m be a nonnegative integer. We consider the generalized cluster complex ∆ m (Φ) defined by S. Fomin and N. Reading and the poset N C (m) (W ) of m-divisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of ∆ m (Φ) in terms of N C (m) (W ), generalizing that of T. Brady and C. Watt given in the case m = 1. Making use of this, we give a case free proof of a conjecture of F. Chapoton and D. Armstrong, which relates a certain refined face count of ∆ m (Φ) with the Möbius function of N C (m) (W ).
Let W be a Weyl group corresponding to the root system A n-1 or Bn. We define a simplicial comple... more Let W be a Weyl group corresponding to the root system A n-1 or Bn. We define a simplicial complex ∆ m W in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆ m W is isomorphic to the cluster complex corresponding to W , defined in . We enumerate the faces of ∆ m W and show that the entries of its h-vector are given by the generalized Narayana numbers N m W (i), defined in . We also prove that for any m ≥ 1 the complex ∆ m W is shellable and hence Cohen-Macaulay.
Let be an irreducible crystallographic root system with Weyl group W and coroot latticeQ, spannin... more Let be an irreducible crystallographic root system with Weyl group W and coroot latticeQ, spanning a Euclidean space V . Let m be a positive integer and A m be the arrangement of hyperplanes in V of the form (α, x) = k for α ∈ and k = 0, 1, . . . , m. It is known that the number N + ( , m) of bounded dominant regions of A m is equal to the number of facets of the positive part m + ( ) of the generalized cluster complex associated to the pair ( , m) by S. Fomin and N. Reading.
Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized clust... more Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex ∆ m (Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that ∆ m (Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part ∆ m + (Φ) of ∆ m (Φ). An explicit homotopy equivalence is given between ∆ m + (Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.
Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a constant number of m distinct ra... more Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a constant number of m distinct radii ρ 1 , ρ 2 , . . . , ρ m , we show that the worst-case combinatorial complexity of the convex hull of Σ is Θ( 1≤i =j≤m n i n ⌊ d 2 ⌋ j ), where n i is the number of spheres in Σ with radius ρ i . To prove the lower bound, we construct a set of Θ(n 1 +n 2 ) spheres in E d , with d ≥ 3 odd, where n i spheres have radius ρ i , i = 1, 2, and ρ 2 = ρ 1 , such that their convex hull has combinatorial complexity Ω(n 1 n
Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used th... more Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.
Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and ... more Consider a set of $r$ convex $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $r\ge{}2$, and let $n_i$ be the number of vertices of $P_i$, $1\le{}i\le{}r$. It has been shown by Fukuda and Weibel that the number of $k$-faces of the Minkowski sum, $P_1+P_2+...+P_r$, is bounded from above by $\Phi_{k+r}(n_1,n_2,...,n_r)$, where $\Phi_{\ell}(n_1,n_2,...,n_r)= \sum_{\substack{1\le{}s_i\le{}n_i s_1+...+s_r=\ell}} \prod_{i=1}^r\binom{n_i}{s_i}$, $\ell\ge{}r$. Fukuda and Weibel have also shown that the upper bound mentioned above is tight for $d\ge{}4$, $2\le{}r\le{}\lfloor\frac{d}{2}\rfloor$, and for all $0\le{}k\le{}\lfloor\frac{d}{2}\rfloor-r$. In this paper we construct a set of $r$ neighborly $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $2\le{}r\le{}d-1$, for which the upper bound of Fukuda and Weibel is attained for all $0\le{}k\le{}\lfloor\frac{d+r-1}{2}\rfloor-r$. Our approach is based on what is known as the Cayley trick for Minkowski sums. A direct consequence of our r...
Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated... more Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Brändén and Solus have given sufficient conditions under which the image of a polynomial under a certain operator associated to barycentric subdivision has such a decomposition. This paper gives a new proof of their result which generalizes to subdivision operators in the setting of uniform triangulations of simplicial complexes, introduced by the first named author. Sufficient conditions under which these decompositions are also interlacing are described. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in f -vector theory arise from this work.
In this paper we present a bijection between two well known families of Catalan objects: the set ... more In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid\left\langle v,\alpha \right\rangle=m\}$ as separating wall. As a result, it restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.
Motivated by the relation N m (C n ) = (mn+1)N m (A n−1 ), holding for the m-generalized Catalan ... more Motivated by the relation N m (C n ) = (mn+1)N m (A n−1 ), holding for the m-generalized Catalan numbers of type A and C, the connection between dominant regions of the m-Shi arrangement of type A n−1 and C n is investigated. More precisely, it is explicitly shown how mn + 1 copies of each element of the set R m + (A n−1 ) of dominant regions of the m-Shi arrangement of type A n−1 , biject onto the set R m + (C n ) of type C n such regions. This is achieved by exploiting two different viewpoints to express the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between mn + 1 copies of each m-Dyck path of height n and the set of N − E lattice paths inside an n × mn rectangle is provided.
We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski su... more We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 + ⋯ + P r , of r convex d-polytopes P 1 , . . . , P r in R d , where d ≥ 2 and r < d, as a (recursively defined) function on the number of vertices of the polytopes. Our results coincide with those recently proved by Adiprasito and Sanyal . In contrast to Adiprasito and Sanyal's approach, which uses tools from Combinatorial Commutative Algebra, our approach is purely geometric and uses basic notions such as f -and h-vector calculus and shellings, and generalizes the methodology used in [15] and [14] for proving upper bounds on the f -vector of the Minkowski sum of two and three convex polytopes, respectively. The key idea behind our approach is to express the Minkowski sum P 1 + ⋯ + P r as a section of the Cayley polytope C of the summands; bounding the k-faces of P 1 + ⋯ + P r reduces to bounding the subset of the (k + r − 1)-faces of C that contain vertices from each of the r polytopes. We end our paper with a sketch of an explicit construction that establishes the tightness of the upper bounds. arXiv:1502.02265v1 [cs.CG] 8 Feb 2015 vertices or number of facets of the summands. Fogel, Halperin, and Weibel [6] extended one of the results in [7], and expressed the number of facets of the Minkowski sum of r 3-polytopes in terms of the number of facets of the summands. Quite recently Weibel [21] provided a relation for the number of k-faces of the Minkowski sum of r ≥ d summands in terms of the k-faces of the Minkowski sums of subsets of size d − 1 of these summands. This result should be viewed in conjunction with a result by Sanyal [19] stating that the number of vertices of the Minkowski sum of r d-polytopes, where r ≥ d, is strictly less than the product of the vertices of the summands (whereas for r ≤ d−1 this is indeed possible). About 3 years ago, the authors of this paper proved the first tight upper bound on the number of k-faces for the Minkowski sum of two d-polytopes in R d , for any d ≥ 2 and for all 0 ≤ k ≤ ), a result which was subsequently extended to three summands in collaboration with Konaxis cf. ).
In this paper we construct a set of $r$ neighborly $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}... more In this paper we construct a set of $r$ neighborly $d$-polytopes $P_1,P_2,...,P_r$, where $d\ge{}3$ and $2\le{}r\le{}d-1$, for which the upper bound of Fukuda and Weibel is attained for all $0\le{}k\le{}\lfloor\frac{d+r-1}{2}\rfloor-r$. Our approach is based on what is known as the Cayley trick for Minkowski sums. A direct consequence of our result is a tight asymptotic bound on the complexity of the Minkowski sum $P_1+P_2+...+P_r$, for any fixed dimension $d$ and any $2\le{}r\le{}d-1$, when the number of vertices of the polytopes is (asymptotically) the same.
In this paper we present a bijection $\omega_n$ between two well known families of Catalan object... more In this paper we present a bijection $\omega_n$ between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and the set of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, $\omega_n$ bijects the facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid<v,\alpha >=m\}$ as separating wall. As a result, $\omega_n$ restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. The map $\omega_n$ is a composition of two bijections in which integer partitions in an $m$-staircase shape come into play.
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, 2012
We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski su... more We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 ⊕ P 2 , of two d-dimensional convex polytopes P 1 and P 2 , as a function of the number of vertices of the polytopes.
We end with algorithmic considerations, and we show how our tight bounds for the parallel polytop... more We end with algorithmic considerations, and we show how our tight bounds for the parallel polytope convex hull problem, yield tight bounds on the combinatorial complexity of the Minkowski sum of two convex polytopes in $\mathbb{E}^d$.
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13, 2013
We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski su... more We derive tight expressions for the maximum number of k-faces, 0 ≤ k ≤ d − 1, of the Minkowski sum, P 1 + P 2 + P 3 , of three d-dimensional convex polytopes P 1 , P 2 and P 3 , as a function of the number of vertices of the polytopes, for any d ≥ 2. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope C, the problem of counting the number of k-faces of P 1 + P 2 + P 3 , reduces to counting the number of (k + 2)-faces of the subset of C comprising of the faces that contain at least one vertex from each P i . In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of r d-polytopes, where r ≥ d. For d ≥ 4, the maximum values are attained when P 1 , P 2 and P 3 are d-polytopes, whose vertex sets are chosen appropriately from three distinct d-dimensional moment-like curves.
Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011
Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a fixed number of m distinct radii... more Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a fixed number of m distinct radii ρ 1 , ρ 2 , . . . , ρ m , we show that the worst-case combinatorial complexity of the convex hull
Let Φ be an finite root system with corresponding reflection group W and let m be a nonnegative i... more Let Φ be an finite root system with corresponding reflection group W and let m be a nonnegative integer. We consider the generalized cluster complex ∆ m (Φ) defined by S. Fomin and N. Reading and the poset N C (m) (W ) of m-divisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of ∆ m (Φ) in terms of N C (m) (W ), generalizing that of T. Brady and C. Watt given in the case m = 1. Making use of this, we give a case free proof of a conjecture of F. Chapoton and D. Armstrong, which relates a certain refined face count of ∆ m (Φ) with the Möbius function of N C (m) (W ).
Let W be a Weyl group corresponding to the root system A n-1 or Bn. We define a simplicial comple... more Let W be a Weyl group corresponding to the root system A n-1 or Bn. We define a simplicial complex ∆ m W in terms of polygon dissections for such a group and any positive integer m. For m = 1, ∆ m W is isomorphic to the cluster complex corresponding to W , defined in . We enumerate the faces of ∆ m W and show that the entries of its h-vector are given by the generalized Narayana numbers N m W (i), defined in . We also prove that for any m ≥ 1 the complex ∆ m W is shellable and hence Cohen-Macaulay.
Let be an irreducible crystallographic root system with Weyl group W and coroot latticeQ, spannin... more Let be an irreducible crystallographic root system with Weyl group W and coroot latticeQ, spanning a Euclidean space V . Let m be a positive integer and A m be the arrangement of hyperplanes in V of the form (α, x) = k for α ∈ and k = 0, 1, . . . , m. It is known that the number N + ( , m) of bounded dominant regions of A m is equal to the number of facets of the positive part m + ( ) of the generalized cluster complex associated to the pair ( , m) by S. Fomin and N. Reading.
Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized clust... more Let Φ be a finite root system of rank n and let m be a nonnegative integer. The generalized cluster complex ∆ m (Φ) was introduced by S. Fomin and N. Reading. It was conjectured by these authors that ∆ m (Φ) is shellable and by V. Reiner that it is (m + 1)-Cohen-Macaulay, in the sense of Baclawski. These statements are proved in this paper. Analogous statements are shown to hold for the positive part ∆ m + (Φ) of ∆ m (Φ). An explicit homotopy equivalence is given between ∆ m + (Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong.
Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a constant number of m distinct ra... more Given a set Σ of spheres in E d , with d ≥ 3 and d odd, having a constant number of m distinct radii ρ 1 , ρ 2 , . . . , ρ m , we show that the worst-case combinatorial complexity of the convex hull of Σ is Θ( 1≤i =j≤m n i n ⌊ d 2 ⌋ j ), where n i is the number of spheres in Σ with radius ρ i . To prove the lower bound, we construct a set of Θ(n 1 +n 2 ) spheres in E d , with d ≥ 3 odd, where n i spheres have radius ρ i , i = 1, 2, and ρ 2 = ρ 1 , such that their convex hull has combinatorial complexity Ω(n 1 n
Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used th... more Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.
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