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Topic: Ferrers graph

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	Integer partition (Site not responding. Last check: 2007-10-12)
		can be obtained by the aid of a visual tool, a Ferrers graph (also called Ferrers diagram, since it is not a graph in the graph-theoretical sense, or sometimes Young diagram, alluding to the Young tableau).
		The graphs for the 5 partitions of the number 4 are
		The number of partitions of a positive Integer n is given by the partition function p(''n'').
	integer-partition.mindbit.com (636 words)

	Combinatorica.nb
		The internals of the graph representation are not shown to the user—only a notation with the number of edges and vertices, followed by whether the graph is directed or undirected.
		Graph theory is the study of properties or invariants of graphs.
		The girth of a graph is the length of its shortest cycle.
	www.cs.sunysb.edu /~skiena/combinatorica/help.html (2168 words)

	Kids.Net.Au - Encyclopedia > Integer partition (Site not responding. Last check: 2007-10-12)
		This and other results can be obtained by the aid of a visual tool, Ferrers' graph.
		The 14 circles are lined up in 4 columns, each having the size of a part of the partition.
		Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
	www.kids.net.au /encyclopedia-wiki/in/Integer_partition (767 words)

	PLOTS AGAINST INFORMATION LAWS (Site not responding. Last check: 2007-10-12)
		In the number theory such diagrams are known as Ferrers graphs and are used as a tool to prove some theorems in the theory of partitions [21].
		This is possible, if we change linear scales of the Ferrers graph by another suitable scale, which weights index j or index k or both indices simultaneously.
		They are different transformations of Ferrers graphs, which change their symmetries and answer different questions.
	mujweb.cz /veda/kunzmilan/plots.htm (2472 words)

	DCI 2001 Research Program Abstracts - Week 1
		Interval graphs and probe interval graphs were introduced for studies in certain biological fields specializing in genetics in the late 50's (for interval graphs), and in the late 90's (for probe interval graphs).
		A probe interval graph is an variation of interval graph that arose from the DNA physical mapping of molecular biology.
		Unit probe interval graph is a special case of probe interval graph where we require that all the intervals assigned to the vertices must have the same length.
	dimacs.rutgers.edu /dci/2001/abstractswk1right.html (2607 words)

	Graph Theory Glossary - co (Site not responding. Last check: 2007-10-12)
		Kahest paarikaupa tippude reast koosnev graaf, kus kõik tipud peale paaristippude on ühendatud omavahel servaga.
		Graph whose vertices are the bridges of a cycle (bridges conflict if they have three common endpoints or four alternating endpoints on the cycle).
		In a directed graph, or network, the path may be required to be directed (i.e., follow the orientations of the arcs), in which case the network is strongly connected; or, the path may be allowed to ignore the arc orientations, in which case the network is weakly connected.
	www.cc.ioc.ee /jus/gtglossary/gtglos_co.htm (4113 words)

	Combinatorica
		This random graph can be expected to have half the number of edges of a complete graph, even though all labeled graphs occur with equal probability.
		The smallest non-trivial self-complementary graphs are the path on four vertices and the cycle on five.
		Planar graphs are graphs which can be embedded in the plane with no pair of edges crossing.
	documents.wolfram.com /v4-de/AddOns/DiscreteMath/Combinatorica.html (1971 words)

	[No title]
		Introduce the Ferrers Graph (Diagram) and use it to show that the number of partitions of n into k parts equals the number of partitions of n in which the largest part equals k.
		Theorem that every graph can be coloured with D+1 colours, where D is the largest degree of a vertex of G, and a sketch of the proof.
		Statement of Brooks' Theorem that any graph which is neither a complete graph nor an odd cycle can be coloured with D colours, where D is the largest degree of a vertex.
	www.math.uvic.ca /faculty/gmacgill/M222F01/222covered.html (2653 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		In a similar spirit, not because it is central to graph theory, but because it is a useful device, graph coloring is deployed throughout the book as a unifying theme.
		Apart from this general prerequisite, the recipe for counting nonisomorphic graphs, found in the latter part of Chapter 10, supposes the reader to be acquainted with the disjoint cycle factorization of a permutation.
		Intended to be neither a comprehensive overview nor an encyclopedic reference, this focused introduction to graph theory goes deeply enough into a sufficiently wide variety of topics to demonstrate its flavor, power, elegance, and vitality.
	www.sci.csuhayward.edu /~rmerris/graphs.html (711 words)

	"Glossary of Terms in Combinatorics"
		Acyclic orientation - an orientation of a graph that is an acyclic digraph
		Grötzsch graph - the smallest triangle-free 4-chromatic graph
		Interval graph - a graph having an interval representation
	www.math.uiuc.edu /~west/openp/gloss.html (11773 words)

	Publications of Jeremy L. Martin
		As another application, we exhibit two infinite families of trees (spiders and some caterpillars), and one family of unicyclic graphs (squids) whose members are determined completely by their chromatic symmetric functions.
		Like its cousin the Erdös-Rényi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected (and in fact has isolated vertices in most cases).
		Abstract: A picture P of a graph G = (V,E) consists of a point P(v) for each vertex v in V and a line P(e) for each edge e in E, all lying in the projective plane over a field k and subject to containment conditions corresponding to incidence in G.
	www.math.ku.edu /~jmartin/pubs.html (1853 words)

	[No title]
		If the graph is edge weighted, then the function returns a matching with maximum total weight." BipartiteMatchingAndCover::usage = "BipartiteMatchingAndCover[g] takes a bipartite graph g and returns a matching with maximum weight along with the dual vertex cover.
		The Petersen graph is identical to the generalized Petersen graph with n = 5 and k = 2." GetEdgeLabels::usage = "GetEdgeLabels[g] returns the list of labels of the edges of g.
		The third item in a Graph object is opts, a sequence of zero or more global options that apply to all vertices or all edges or to the graph as a whole.
	www.cs.uiowa.edu /~sriram/Combinatorica/NewCombinatorica.m (6443 words)

	Partition diagrams (Site not responding. Last check: 2007-10-12)
		A tree structure of Bell number diagrams allows recursive partitioning to be displayed in a less visually confusing manner that displaying it with a standard tree graph.
		Fa`a di Bruno's Formula uses the enumeration of the integer partitions to form the index for the summation that gives the different derivatives of composite functions.
		The Ferrers diagrams below visually enumerate the integer partitions of the number four.
	www.tetration.org /Combinatorics/PartitionDiagrams.htm (569 words)

	Partition (number theory) - Wikipedia, the free encyclopedia
		The 14 circles are lined up in 4 columns, each having the size of a part of the partition.
		By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14.
		Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.
	en.wikipedia.org /wiki/Partition_(number_theory) (1501 words)

	Abstract (Site not responding. Last check: 2007-10-12)
		To a Ferrers tableau we associate a bipartite graph.
		It turns out that all Ferrers ideals have a linear minimal free resolution which we completely describe, including the maps.
		We also determine the primary decomposition of the Ferrers ideals and show that a Ferrers ideal is unmixed if and only if it is Cohen-Macaulay.
	www.math.tamu.edu /research/algcom/history_05-06/060217a.html (86 words)

	arxivmath: Monomial and toric ideals associated to Ferrers graphs. [math.AC/0609371]
		Its edge ideal, dubbed Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics.
		In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs.
		Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution: This is obtained by associating a suitable polyhedral cell complex to the ideal/graph.
	arxivmath.livejournal.com /1589062.html (206 words)

	October 1998 Seminars (Site not responding. Last check: 2007-10-12)
		Abstract: We consider the problem of representing integers as sums of powers of 2, each part appearing at most twice, obtaining analytic and algebraic expressions for the number of representations of n, the asymptotics for the summatory function, and connections with other areas of combinatorics, such as Stern-Brocot trees.
		Abstract: Kostochka conjectures that for every integer t there exists a constant c=c(t) such that for every graph G on n vertices, either G can be contracted onto the complete graph on t+1 vertices, or the complement of G can be contracted onto a complete graph on at least (1+1/t)n/2-c vertices.
		We prove that Kostochka's conjecture is equivalent to the assertion that every graph that cannot be contracted onto the complete graph on t+1 vertices has an independe nt set of size at least n/t, and deduce that the conjecture holds for t
	www.math.gatech.edu /aco/plain/events/old/csem/oct98.html (307 words)

	Relatively recent papers- Vic Reiner
		He observed that the Schubert cell structure of such a variety is indexed by maximal rook placements on the Ferrers board, and that the integral cohomology groups of two such varieties are additively isomorphic exactly when the Ferrers boards satisfy the combinatorial condition of rook-equivalence.
		ABSTRACT: The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian.
		For Cartesian products of complete graphs, we generalize results of H. Bai on the k-dimensional cube, by bounding the number of invariant factors in the critical group, and describing completely its p-primary structure for all primes p that divide none of the sizes of the complete graph factors.
	www.math.umn.edu /~reiner/Papers/papers.html (4505 words)

	Foundations of Combinatorics
		Graph Theory: Graphs are among the most important structures in Combinatorics.
		The existence questions of Combinatorics are prevalent in graph theory.
		Symmetric Functions: A symmetric function is a power series in countably many variables which is invariant under transposing any two variables.
	www.math.washington.edu /~billey/classes/581.html (757 words)

	[No title]
		Extremal graph theory: Turan's theorem, the Erdos-Gallai theorem.
		An introduction to the concepts of directed graphs, problems about directed circuits and directed cuts, and minimax equalities relating these and other graphical objects.
		Included are algorithms for graph isomorphism, planarity, connectedness, chromatic number, spanning trees, circuits and many others.
	www.ucalendar.uwaterloo.ca /SA/GRAD/test/GRDcourse-CO.html (995 words)

	Graphical Basis Partitions (ResearchIndex) (Site not responding. Last check: 2007-10-12)
		Abstract: A partition of an integer n is graphical if it is the degree sequence of a simple, undirected graph.
		It is an open question whether the fraction of partitions of n which are graphical approaches 0 as n approaches infinity.
		A partition is basic if the number of dots in its Ferrers graph is minimum among all partitions with the same rank vector as.
	citeseer.ist.psu.edu /295984.html (381 words)

	mike develin: the works
		My current research focus is in combinatorial geometry, specifically polytopes; if it weren't, it might be in graph theory, combinatorics, or discrete math in general.
		A general notion of visibility graphs, with S.
		On the bondage number of planar and directed graphs, 2003, with K. Carlson, Discrete Math, 306 (2006), pp.
	math.berkeley.edu /~develin (702 words)

	CSUEB Catalog 2006-2007: Mathematics (Undergraduate)
		Operations with algebraic expressions, exponents and radicals; linear and quadratic equations; systems of equations and inequalities; linear and quadratic functions and their graphs; elementary conic sections; word problems.
		Introduction to methods and proof techniques in several branches of mathematics, with topics chosen from logic, set theory, abstract algebra, number theory, analysis, and graph theory.
		Planar graphs and the theorems of Euler and Kuratowski.
	www.csueastbay.com /ecat/20062007/u-math.html (3051 words)

	Marko Riedel's combinatorics and number theory page (Site not responding. Last check: 2007-10-12)
		With cycle indices for k-partite cycles and the bow-tie graph.
		Expected size of the automorphism group of a random BST.
		Cover time: any graph G, any vertex v.
	www.geocities.com /markoriedelde/combnumth.html (1137 words)

	[No title]
		It is a concise treatment of the aspects of intersection graphs that interconnect many standard concepts and form the foundation of a surprising array of applications to biology, computing, psychology, matrices, and statistics.
		Some of the applications are not widely known or available in the graph theoretic literature and are presented here for the first time.
		This book is written for students who have completed a basic graph theory course and for mathematicians seeking a guide for graph theory and graph modeling.
	www.ec-securehost.com /SIAM/DT02.html (370 words)

	Seminar for Women in Mathematics (Site not responding. Last check: 2007-10-12)
		We'll look at two elementary methods for analyzing partitions: Ferrers graphs and generating functions, and then briefly discuss how the theory of modular forms has led to some recent surprising results about p(n).
		The distinguishing chromatic number of a graph G is the least integer k such that there is a proper k-coloring of G which is not preserved by any nontrivial automorphism of G.
		We will show that the distinguishing chromatic number of G^d (the Cartesian product of G by d times) is at most one more than the usual chromatic number of G for d at least 6, where G is either a complete graph or a hypercube.
	www.math.uiuc.edu /~jarobin1/WomensSeminar.html (416 words)

	Pentagonal number theorem - Wikipedia, the free encyclopedia
		Let k be the number of elements in the smallest row of our graph.
		In our case, this takes us back into the first graph.
		which fails to change the parity of the number of rows, and is not reversible in the sense that performing the operation again does not take us back to the original graph.
	en.wikipedia.org /wiki/Pentagonal_number_theorem (954 words)

	Science and Technology News - June 17th, 2006
		It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees.
		We determine the arithmetical rank of every edge ideal of a Ferrers graph.
		We show that a proalgebraic version of the Euler--Poincar\'e characteristic with values in the Grothendieck ring is a generalization of the so-called motivic measure.
	vsevcosmos.livejournal.com /2006/06/17 (6259 words)

	The Combinatorics Seminar at Chalmers/GU
		Let G=(V,E) be a simple undirected graph with vertex set V={1,2,..., n} and edge set E. An independent set (also called a stable set) of G is a subset S of V such that no two vertices in S are have an edge between them.
		For instance, the set of independent sets of a staircase Ferrers graph on 2n vertices is in one-to-one correspondence with the parts of all compositions (ordered non-empty partitions) of n+1.
		Bodo Lass (Lyon): Acyclic orientations and the chromatic polynomial of a graph
	www.math.chalmers.se /Math/Research/Combinatorics/kombseminar.html (8722 words)

	A History of the Combinatorial Potlatches
		The edge-toughness of a graph and of its complement
		Graph theory as an integral part of mathematics
		BA: You will note that Richard Weiss is listed as giving the same talk at two consecutive potlatches.
	buzzard.ups.edu /potlatch/history.html (509 words)

	Front: [math.AC/0606353] A note on the edge ideals of Ferrers graphs (Site not responding. Last check: 2007-10-12)
		Front: [math.AC/0606353] A note on the edge ideals of Ferrers graphs
		Title: A note on the edge ideals of Ferrers graphs
		Abstract: We determine the arithmetical rank of every edge ideal of a Ferrers graph.
	front.math.ucdavis.edu /math.AC/0606353 (79 words)

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