# Questions tagged [planar-partitions]

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11
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### 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
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### 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,...

9
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1
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### 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

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### 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
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0
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### 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 ...

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0
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### 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 ...

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0
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### 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,...

3
votes

1
answer

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### 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 ...

2
votes

2
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### 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 ...

2
votes

0
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### 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

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### 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 ...