All Questions
Tagged with lattices co.combinatorics
67 questions
7
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1
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1k
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On "The Average Height of Planted Plane Trees" by Knuth, de Bruijn and Rice (1972)
I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
4
votes
2
answers
323
views
Cancellation theorem for lattices
By a lattice, we mean a finitely generated, free $\mathbb{Z}$-module together with a symmetric bilinear form. Typical examples are the hyperbolic lattices $U$ and the root lattices $A_{n}, D_{n}, E_{n}...
- 41
2
votes
1
answer
267
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Expected number of identical vertex pairs with the same Euclidean distance on a randomly colored rectangular lattice
Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the ${N*M}\choose{2}$ possible unordered pairs ...
- 23
16
votes
4
answers
597
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The lattice spanned by $m$ random 0-1 vectors of length $n$
Consider $m$ random 0-1 vectors of length $n$. Let $L$ be the lattice spanned by them. What is the value of $m$ (as a function of $n$) for which it is true with positive probability that $L=Z^n$? More ...
- 24.7k
2
votes
1
answer
300
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Spanning set for Lattice generated by an orbit of the group.
For a vector spaces it always holds that any set of vectors spanning vector space $V$ has a subset of vectors which is a basis for $V$. While for lattices it is not true. For example consider one ...
- 2,179
1
vote
3
answers
470
views
Triangulations of lattice polygons
Let L be a 2-dimensional lattice and P- a lattice polygon. Suppose, it is triangulated into lattice tiangles. What are restrictions on their areas? For instance, can a lattice triangle of even area ...
- 4,316
0
votes
0
answers
113
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Question regarding contiguous forms
I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...
- 149
7
votes
2
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963
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Maximal number of edges and triangular cells for n points in a triangular lattice
Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...
- 567
6
votes
1
answer
451
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Orthogonal Complements of Root Lattices in E_8
I have a rather stupid lattice theory question. Suppose $L$ is a root lattice that can be primitively embedded in the $ E_8 $ lattice. Is the orthogonal complement of $ L$ in $E_8$ unique up to ...
- 309
4
votes
0
answers
242
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Domination in Nice Lattices
Let an integer vector be nice when it has only two nonzero components, which sum to zero. So (0, 0, 3, 0, -3) and (-1, 0, 1, 0, 0) are examples of nice vectors in $n=5$ dimensions.
Call a lattice ...
- 1,288
2
votes
1
answer
337
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Count of lattices on finite set
Let $p(n)$ denote count of lattices on finite set $G$, $|G|=n$ (without isomorphism). It's know closed formula for $p(n)$?
It's clear, that $1 \leq p(n)$ and also that $p(n-1) \leq p(n)$ for $n \geq ...
- 125
0
votes
1
answer
241
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Constructing a smooth lattice from a discrete one.
I have the standard lattice L defined over partitions of $1\ldots n$ under the split-merge relation. I also have an antimonotone function from L to R that's submodular, and so gives me a metric on L ...
- 4,462
10
votes
1
answer
595
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Condition for existence of certain lattice points on polytopes
Let $a_1,\cdots, a_n$ be integers such that $a_i\geq 2$ for all $i$ and $k>0$ another integer.
I am interested in whether there exist integers $x_1,\cdots, x_n$ with $0<x_i<a_i$ satisfying:
...
- 30.5k
0
votes
2
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611
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10
votes
1
answer
803
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Which lattices have more than one minimal periodic coloring?
The lattice $\mathbb{Z}^n$ has an essentially unique (up to permutation) minimal periodic coloring for all $n$, namely the "checkerboard" 2-coloring. Here a coloring of a lattice $L$ is a coloring of ...
- 15.4k
22
votes
4
answers
2k
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What exactly is the relationship between codes over finite fields and Euclidean sphere-packings?
So I know that error-correcting codes are sphere packings in the Hamming metric, and that intuition and technical tools from the Euclidean case can often be applied to the finite-field case and vice ...
- 12.6k
12
votes
3
answers
707
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A "round" lattice with low kissing number?
Historically, the lattices with high density were studied intensively, e.g. E_8 lattice or Leech Lattice. However, there are situations that lattices with low kissing number are required. Specifically,...
- 139