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No two blocks "cross" each other
- A noncrossing partition of S is a partition in which no two blocks "cross" each other, i.e., if a and b belong to one block and x and y to another, they are not arranged in the order a x b y.
en.wikipedia.org/wiki/Noncrossing_partition
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A noncrossing partition of S is a partition in which no two blocks "cross" each other, i.e., if a and b belong to one block and x and y to another, they are not arranged in the order a x b y.
A non-crossing partition is a partition of the vertices of a regular n-gon (labeled by the σ set [n ]) with the property that the convex hulls of its blocks are pairwise disjoint. Fig-ure 4 illustrates the noncrossing partition {{1 , 4 5 , }, {2 , 3 }, {6 , 8 }, {7}}.
A partition of n (i.e. of [n]) is called noncrossing if there do not exist integers 1 ≤ a < b < c < d ≤ n so that a,c lie in one part and b,d lie in another part.
Jul 6, 2010 · Non-crossing partitions of an ordered set. D efinitions 9.1. Let S be a finite totally ordered set. (1) We call π = {V1, …, Vr} a partition of the set S if and only if the Vi (1 ≤ i ≤ r) are pairwise disjoint, non-void subsets of S such that V1 ∪ … ∪ Vr = S. We call V1, …, Vr the blocks of π. The number of blocks of π is denoted by |π|.
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation.
A non-crossing partition is a partition in which lines joining members of the same subset do not intersect any lines joining members of another subset. One representation of noncrossing partitions is one in which points are arranged at the corners of a regular polygon, with line segments connecting members of the same block.
formally: Given a noncrossing partition ˇof [n], draw its linear representation L, which either (1) contains the arc (i;j) or (2) does not. In case (1), let L0be Lwith arc (i;j) removed; in case (2), let L 0be Lwith arc (i;j) added. If L is the linear representation of some noncrossing partition ˇ0(guaranteed to exist in case (1) but not ...
