Questions tagged [planar-partitions]

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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 ...
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0answers
90 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 ...
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52 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 ...
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27 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,...
3
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1answer
154 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 ...
2
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2answers
134 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 ...
2
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0answers
80 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
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1answer
384 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 ...