Questions tagged [planar-partitions]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
12 votes
2 answers
448 views

Generating function for counting partitions with corners

A corner of an integer partition is a location at where a box can be added to its Ferrers diagram to give a new partition. E.g. the partition $\{1,1,1\}$ has two corners, and $\{1,2\}$ has three ...
1 vote
0 answers
30 views

What are some other methods for partitioning an n-dimensional space based on a set of points in that space?

So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...
  • 11
9 votes
1 answer
183 views

Bi-partitioning $2n$ points on the plane with a straight line

Let $S$ be a set of $2n$ points in $\mathbb{R}^2$. Which is the maximum number of different bi-partitions of $S$ generated by a straight line? More precisely, which is the maximum number of partitions ...
2 votes
1 answer
230 views

Objects in bijection with integer partitions (and lattices)

A partition of $n$ is a non-increasing sequence of positive integers of sum $n$. Several lattices are defined over integer partitions, in particular the dominance order and the Young lattice. Several ...
2 votes
0 answers
96 views

Lengths of cycles in non-crossing partitions

Let $g$ be an element of the permutation group $S_n$ and let $\eta$ be the cyclic permutation $(1,2,\dots,n)$. Define $D(g)$ as the number of cycles in permutation $g$. I am aware of the fact that the ...
1 vote
0 answers
54 views

Convex Triangulations II

While the original question Convex triangulations was aimed at the existence and calculation of convex triangulations of a given set of $n$ points in the Euclidean plane, I would like to ask the ...
  • 11.3k
1 vote
0 answers
32 views

Definition of convex hulls via maximal sets of interior-disjoint simplices

Let the simplex cover of a finite set $\mathcal{P}\subset \mathbb{E}^n,\ n\,\le\, k:=\operatorname{card}(\mathcal{P})\,\lt\infty$ of points in Euclidean $n$-space of which no $n+1$ are co-hyperplanar,...
  • 11.3k
3 votes
1 answer
159 views

Convex triangulations

Given a set of $n$ points in the Euclidean plane of which no three are collinear, does there always exist a convex triangulation and how can one be found algorithmically? In this context a convex ...
  • 11.3k
2 votes
2 answers
151 views

enumerate line partitions of points in the plane

Let $S$ be a nonempty subset of $\mathbb{R}^2$ and $l$ a line in $\mathbb{R}^2$ disjoint from $S$. Then $l$ partitions $S$ into two disjoint sets $S = S_1 \cup S_2$ in the obvious way. Note that, at ...
  • 171
2 votes
0 answers
85 views

8-partition of a planar convex body by 4 concurrent lines

It is known1 that any convex body $K$ in the plane can be partitioned into $6$ equal-area pieces by $3$ concurrent lines which meet at a point in $K$. Call this a $6$-partition. This result cannot be ...
13 votes
1 answer
404 views

Counting higher-dimensional partitions with symmetric function theory

My coauthors and I are writing a (mostly expository) paper in which we construct the Specht module. Our proof that the Specht module is irreducible in characteristic zero implies the following ...