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	Partition of a set - Wikipedia, the free encyclopedia
		In mathematics, a partition of a set X is a division of X into non-overlapping "parts" or "blocks" or "cells" that cover all of X.
		A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets.
		The lattice of noncrossing partitions of a finite set has recently taken on importance because of its role in free probability theory.
	en.wikipedia.org /wiki/Partition_of_a_set (734 words)

	Partition of a set -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A partition of a set X is a set of (Click link for more info and facts about nonempty) nonempty (A set whose members are members of another set; a set contained within another set) subsets of X such that every element x in X is in exactly one of these subsets.
		is not a partition of because none of its blocks contains 3; however, it is a partition of.
		The lattice of (Click link for more info and facts about noncrossing partition) noncrossing partitions of a finite set has recently taken on importance because of its role in (Click link for more info and facts about free probability) free probability theory.
	www.absoluteastronomy.com /encyclopedia/p/pa/partition_of_a_set.htm (809 words)

	CS267: Notes for Lectures 20 and 21, Mar 21 1996 and Apr 2 1996
		Graph partitioning, which was introduced in Lecture 16, is a technique for executing a set of tasks in parallel so as to balance the load and minimize communications among processors.
		Graph partitioning can be used to reorder the rows and columns of the matrix in such as way as to dramatically decrease the number of nonzero entries created during elimination, and the number of floating point operations required.
		Partition the graph using H to find a vertex separator as follows: For each node n, let B(n) be a ball centered at n and with radius equal to the distance to the farthest node to which it is connected.
	http.cs.berkeley.edu /~demmel/cs267/lecture18/lecture18.html (5206 words)

	Noncrossing partition -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		A (Click link for more info and facts about partition of a set) partition of a set S is a pairwise disjoint set of non-empty subsets, called "parts" or "blocks", whose union is all of S.
		In the latter two orders the partition is noncrossing.
		The set of all noncrossing partitions is one of many sets enumerated by the (Click link for more info and facts about Catalan number) Catalan numbers, i.e., the number of noncrossing partitions of a set of size n is
	www.absoluteastronomy.com /encyclopedia/n/no/noncrossing_partition.htm (439 words)

	List of partition topics - Wikipedia, the free encyclopedia
		For the political sense of the word partition see for example: history of Cyprus, history of Ireland, partition of India, partitions of Poland, 1947 UN Partition Plan (Palestine).
		The partition disambiguation page lists meanings in other fields as well.
		a partition of the sum of squares in statistics problems, especially in the analysis of variance.
	en.wikipedia.org /wiki/List_of_partition_topics (123 words)

	Abstract (Site not responding. Last check: 2007-10-08)
		Classical noncrossing partitions are partitions p such that a certain planar diagram of p has non-intersecting parts.
		One obtains a "noncrossing partition lattice" for W which is instrumental in constructing finite K(pi,1) spaces and monoid structures for the Artin group associated to W.
		We also show how the planar diagrams for classical noncrossing partitions and their counterparts of types B and D arise from a uniform construction, rather than in an ad hoc manner and discuss this construction in the case of the exceptional groups.
	www.math.tamu.edu /research/algcom/history_04-05/041029.html (182 words)

	Papers and preprints, etc. (Site not responding. Last check: 2007-10-08)
		Applies this to the poset of partitions of a multiset to show under certain conditions, that it is homotopy equivalent to a wedge of spheres of top dimension.
		Gives a partitioning for the quotient of the partition lattice by the symmetric group, yielding a combinatorial interpretation for the multiplicity of the trivial representation in the rank-selected homology of the partition lattice.
		(with Robert Kleinberg) The refinement complex of the poset of partitions of a multiset.
	www.math.lsa.umich.edu /~plhersh/papers.html (1412 words)

	[No title]
		The study of noncrossing partitions goes back at least to Becker \cite{b}, where they are called ``planar rhyme schemes.'' The systematic study of noncrossing partitions began with Kreweras \cite{k1} and Poupard \cite{p}.
		Let $f(n_1,n_2,\cdots,n_p)$ denote the number of noncrossing partitions of $[n]$ into $p$ parts of given sizes $n_1,n_2,\cdots,n_p$ (but not specifying which part gets which size); and let $p_k$ denote the number of parts with size $k$.
		Then there is a 1-1 onto mapping between the set $\cal N$ of noncrossing partitions of $[n]$ into $p$ parts with given sizes $n_1,n_2,\cdots,n_p$ and the set $\cal V$ of vectors $(k_1, k_2, \cdots, k_{p-1})$ where $1 \le k_i \le n$ and the $k_i$'s are distinct for $1 \le i \le p-1$.
	www.ams.org /proc/1998-126-06/S0002-9939-98-04546-8/S0002-9939-98-04546-8.tex (1100 words)

	[No title]
		It follows that the number of nonnesting partitions on $\Phi$ is equal to the number of regions of the $\Phi$-Catalan arrangement, divided by the order of the group $W$.
		However, the set of nonnesting partitions of $[n]$ partially ordered by refinement will typically not be a lattice, as is the case for noncrossing partitions.
		For example, there are $3$ noncrossing $D_3$-partitions of type $(3)$ but $4$ nonnesting partitions of the same kind, namely the $3$ nonnesting partitions on $B_3$ of type $(3)$ and the one with blocks $\{1, -2, -3\}$ and $\{2, 3, -1\}$.
	www.combinatorics.org /Volume_5/Texfiles/v5i1r42.tex (799 words)

	Non-Crossing Partitions For Classical Reflection Groups - Reiner (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: We introduce analogues of the lattice of non-crossing set partitions for the classical reflection groups of type B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved.
		6: the structure of the lattice of non-crossing partitions (context) - Simion, Ullman - 1991
		24 the structure of the lattice of non-crossing partitions (context) - Simion, Ullman - 1991
	citeseer.ist.psu.edu /reiner96noncrossing.html (674 words)

	Welcome to Adobe GoLive 6
		For instance, the partition {1,2,4,5}{3} is noncrossing while the partition {1,2,4},{3,5} is crossing.
		Noncrossing partitions are naturally associated to certain hyperplane arrangements, certain ideals in a Lie algebra, and certain special subvarieties of the flag variety.
		Partition bijections is a combinatorial approach which often gives the shortest and the most elegant proofs of these identities.
	kcollins.web.wesleyan.edu /feb26program.htm (572 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		,n} is a partition p of the set {1,2,...
		For example, [1,3,7], [2], [4,5], [6], [8,9] is a non crossing set partition, but [1,3], [2,4] is not (take x=1, y=3, z=2, t=4).
		Non crossing set partitions implements the Biane bijection with Dyck words.
	www.sciface.com /STATIC/DOC30/eng/combinat_nonCrossingPartitions.html (309 words)

	[No title]
		A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given.
		The lattice of noncrossing partitions, which was defined first for symmetric groups by G. Kreweras \cite{kreweras}, has been recently generalized to all finite Coxeter groups \cite{bessis,biane,bradywatt}.
		It is probably possible to prove it for classical types using the combinatorial descriptions of the noncrossing partition lattices \cite{athareiner,reiner} and of the generalized associahedra \cite{ysystem}.
	emis.library.cornell.edu /journals/SLC/wpapers/s51chapoton.tex/s51chapoton.tex (4337 words)

	DIMACS Workshop on Geometric Graph Theory
		A collection of pairwise noncrossing simple closed curves in S is a blockage if every onesided simple closed curve in S crosses at least one of them.
		Robertson and Thomas conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is ``roughly'' equal to one half of the minimum number of intersections of a blockage with the graph.
		Combining these results with Sz\'ekely's technique, which is based on crossing numbers of graphs, and with dual space partitioning techniques, we obtain improved bounds for incidences between points and circles, between points and parabolas, between points and pseudo-circles, and between points and graphs of polynomials of fixed maximum degree.
	dimacs.rutgers.edu /Workshops/GeometricGraph/abstracts.html (5604 words)

	Frequently Asked Questions: comp.graphics.algorithms
		This indeed works, but there is a simpler method: the triangulation need not be a partition, but rather can use positively and negatively oriented triangles (with positive and negative areas), as is used when computing the area of a polygon.
		The boundary behavior is complex but determined; in particular, for a partition of a region into polygons, each point is "in" exactly one polygon.
		Usually one restricts corners of the triangles to coincide with vertices of the polygon, in which case every polygon of n vertices can be triangulated, and all triangulations contain n-2 triangles, employing n-3 "diagonals" (chords between vertices that otherwise do not touch the boundary of the polygon).
	femto.cs.uiuc.edu /faqs/cga-faq.html (14000 words)

	FPSAC01: Abstract for Paper 6 (Site not responding. Last check: 2007-10-08)
		We introduce two partially ordered sets, $P^A_n$ and $P^B_n$, of the same cardinalities as the type-A and type-B noncrossing partition lattices.
		In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions.
		We also discuss posets $Q^A_n$ and $Q^B_n$ similarly associated with noncrossing partitions.
	math.la.asu.edu /~fpsac01/PROGRAM/6.html (141 words)

	Selected Math Reviews of Rodica Simion's Papers From MathSciNet
		The form of the expressions for the volumes suggests that each region can itself be partitioned into images of simplices under a measure-preserving transformation, where the simplices have equal volume and their number is the Eulerian number corresponding to that region.
		This condition is that $l(n)$ be odd, where $l(n)$ is the number of partitions of $n$ having an odd number of even summands.
		A partition $a$ dominates $b$ if, with both partitions written as sequences of integers in decreasing order, the partial sums of $a$ are at least as big as the corresponding partial sums of $b$.
	www.math.rutgers.edu /~zeilberg/simion/rsmr.html (1848 words)

	Abstracts (Site not responding. Last check: 2007-10-08)
		The asymptotic methods and the shape of 38 of the 40 identities suggest the influence of the 5-dissection of the generating function for the crank of partitions.
		Using a bijection between partitions and vacillating tableaus, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution.
		As a corollary, the number of $k$-noncrossing partitions is equal to the number of $k$-nonnesting partitions.
	www.theoryofnumbers.com /CANT/2005/abstracts.htm (4450 words)

	Noncrossing Partitions for the Group D_n
		It is a self-dual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2,...,n} defined by Kreweras in 1972 when W is the symmetric group S
		We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B, and C.
		This leads to a (case-by-case) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to W-orbits.
	epubs.siam.org /sam-bin/dbq/article/43219 (230 words)

	Cumulant (Site not responding. Last check: 2007-10-08)
		The coefficient in each term is the number of partitions of a set of n members that collapse to that partition of the integer n when the members of the set become indistinguishable.
		In combinatorics, the nth Bell number is the number of partitions of a set of size n.
		The pattern is that the numbers of blocks in the aforementioned partitions are the exponents on x.
	www.worldhistory.com /wiki/C/Cumulant.htm (1275 words)

	Partición de Noncrossing (Site not responding. Last check: 2007-10-08)
		En matemáticas combinatorias, el asunto de particiones noncrossing ha asumido una cierta importancia debido a (entre otras cosas) su aplicación de la teoría a la probabilidad libremente.
		Una partición noncrossing de S es una partición en la cual los bloques de no dos "cruzan", es decir, si a y b pertenecen a una bloque y x y y a otra, ellos no se arreglan en la orden un x b y.
		El enrejado de noncrossing reparte juegos el mismo papel en definir "cumulants libres" en la teoría de las probabilidades libre que es jugada por el enrejado de todas las particiones en definir cumulants comunes en teoría de las probabilidades clásica.
	www.yotor.net /wiki/es/pa/Partici%F3n%20de%20Noncrossing.htm (553 words)

	Publications and other projects (Site not responding. Last check: 2007-10-08)
		Both can also be indexed by the breadth first ordering of vertices in the non-order contractible planar trees in which precisely one non-degenerate vertex occurs on each level.
		Abstract: We show that for stochastic measures with freely independent increments, the partition-dependent stochastic measures of [Ans00] can be expressed purely in terms of the higher stochastic measures and the higher diagonal measures of the original.
		Abstract: We consider free multiple stochastic measures in the combinatorial framework of the lattice of all diagonals of an n-dimensional space.
	math.ucr.edu /~manshel/pop.html (1677 words)

	Chen's paper - Motzkin paths and reduced decompositions for permutations with forbidden patterns (Site not responding. Last check: 2007-10-08)
		We show that this algorithm preserves the noncrossing property.This yields a simple explanation of an identity due to Simion-Ullman andKlazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures.
		For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers.
		Partition, noncrossing partition, m-regular partition, RNA secondary structure, Davenport-Schinzel sequence, Narayana number, Catalan number.
	www.billchen.org /publications/reduction-m/reduction-m.htm (126 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		A {\sl noncrossing partition} of the set $[n]=\{1,2,\dots,n\}$ is a partition $\pi$ of the set $[n]$ with the property that if $a
		A generating function for the flag $f$-vector of the lattice NC$_{n+1}$ of noncrossing partitions of $[{\scriptstyle n+1}]$ is shown to coincide (up to the involution $\omega$ on symmetric function) with Haiman's parking function symmetric function.
		We construct an edge labeling of NC$_{n+1}$ whose chain labels are the set of all parking functions of length $n$.
	www.combinatorics.org /Volume_4/Abstracts/v4i2r20.tex (124 words)

	Preprints of C.A. Athanasiadis (Site not responding. Last check: 2007-10-08)
		On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements (with E.
		Shellability of noncrossing partition lattices, (with T. Brady and C. Watt)
		Noncrossing partitions for the group Dn (with V.
	www.math.uoc.gr /~caa/preprints.html (135 words)

	Graduate Seminar (Site not responding. Last check: 2007-10-08)
		Chances are you won't get a manifold. We will explore ideas on how to count when a pseudo-manifold actually is a manifold. Some ideas that come up relate to the probability of a k-regular graph being connected, and what is the probability of getting a sphere as a 2-manifold from arbitrary triangulations.
		abstract: A partition of a set S is a collection of subsets of S that are pairwise disjoint and whose union is S. The subsets in the collection are called blocks.
		= 1/(n+1) * binom(2n,n)) and to understand the action of the dihedral group of order 2n on the set of all noncrossing partitions of [n].
	www.math.byu.edu /Seminars/graduate.html (791 words)

	Atlas: Shellability of noncrossing partition lattices by Christos A. Athanasiadis (Site not responding. Last check: 2007-10-08)
		We give a case-free proof that the lattice of noncrossing partitions associated to any finite real reflection group is EL-shellable.
		Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqd-39.
	atlas-conferences.com /cgi-bin/abstract/caqd-39 (90 words)

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