All Questions
7 questions
4
votes
0
answers
287
views
How can we prove this combinatorial identity?
This is a follow up on my earlier MO post. Let's recall the sets
$$\mathbf{K}_n=\{\mathbf{k}\in\mathbb{Z}^n: k_i\geq0, k_1+\cdots+ k_n=n, k_1+\cdots k_i\leq i, \text{for all $1\leq i\leq n$}\}$$
and $\...
- 43.2k
5
votes
1
answer
398
views
Catalan sequences vs composition sequences
In the paper, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Pitman and Stanley studied the $n$-dimensional polytope
$$\Pi_n(\mathbf x)=\{y\in\...
- 43.2k
3
votes
0
answers
154
views
Ehrhart-Macdonald reciprocity with multiplicities
Let $P$ be a convex lattice polytope in $\mathbb{R}^n$. The function $L(t, P) = |\mathbb{Z}^n \cap t\cdot P|$ is a polynomial, and we have an equality
$$L(-t, P) = (-1)^nL(t, P^{int}),$$
where $P^{int}...
- 645
3
votes
0
answers
70
views
On the proportion of simplicial $d$-polytopes on $n$-vertices
I have a question regarding estimates for the proportion of simplicial $d$-polytopes on $n$-vertices.
Let $c_s(n,d)$ denote the number of combinatorial types of simplicial $d$-polytopes on $n$ ...
2
votes
1
answer
340
views
Bit complexity of Barvinok's algorithm
I have seen many references which state Barvinok's algorithm has polynomial time complexity for counting integer points of polytopes in fixed dimension.
What exactly is this arithmetic complexity?
...
- 13.9k
4
votes
1
answer
172
views
The number of simplicial and general $d$-polytopes with $d+3$ labelled vertices
Micha Perles used Gale diagrams to compute the number of simplicial $d$-polytopes with $d+3$ vertices and of general $d$-polytopes with $d+3$ vertices. The computation can be found in Chapter 6.3 of ...
- 24.7k
8
votes
1
answer
725
views
Number of simplicial polytopes with a given f-vector
Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ...
- 902