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Topic: Combinatorial proof

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	Mathematical proof - Wikipedia, the free encyclopedia
		Proof by induction is where a "base case" is proved, and an "induction rule" used to prove an (often infinite) series of other cases.
		Proof by construction, or 'proof by example, is the constructing a concrete example with a property to show that something having that property exists.
		Proof by exhaustion is where the conclusion is established by dividing it into a finite number of cases and proving each one separately.
	en.wikipedia.org /wiki/Mathematical_proof (796 words)

	NationMaster - Encyclopedia: Combinatorics (Site not responding. Last check: 2007-10-31)
		Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties.
		In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence (hence independence structure) that generalizes linear independence in vector spaces.
		In combinatorial mathematics, a matroid is a structure that captures the essence of a notion of independence that generalizes linear independence in vector spaces.
	www.nationmaster.com /encyclopedia/Combinatorics (5554 words)

	Combinatorial proof - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-31)
		A combinatorial proof is a method of proving a statement, usually a combinatorics identity, by counting some carefully chosen object in different ways to obtain different expressions in the statement (see also double counting).
		A statement is said to be proven combinatorially if a combinatorial argument, or counting argument, is used in the aforementioned fashion to justify the key steps of its proof.
		Problems that admit combinatorial proofs are not limited to binomial coefficient identities.
	www.encyclopedia-online.info /Combinatorial_proof (402 words)

	Read This: Proofs That Really Count
		Proofs That Really Count: The Art of Combinatorial Proof, the new book co-written by Benjamin and Jennifer Quinn, is full of exactly this kind of problem and solution.
		While many combinatorialists may find it clear why this type of fact is interesting — or believe that it is interesting merely for the fact that it is true — the fact that this is not clear to me makes me think that it would be even less clear to a student.
		Proofs That Really Count: The Art of Combinatorial Proof, by Arthur T. Benjamin and Jennifer J. Quinn.
	www.maa.org /reviews/reallycount.html (1110 words)

	Amazon.ca: Discrete Math with Proof: Books: Eric Gossett (Site not responding. Last check: 2007-10-31)
		The proof techniques are extensively illustrated in the rest of the text.
		Combinatorial proof is introduced in Chapter 5 and used in Chapter 8 to establish the necessary conditions for the existence of a balanced incomplete block design.
		Many of the more difficult proofs are accompanied by illustrative examples that can be read in parallel with the proof.
	www.amazon.ca /Discrete-Math-Proof-Eric-Gossett/dp/0130669482 (4234 words)

	David Bressoud
		Combinatorial Theory, A. Unimodality of Gaussian polynomials, Discrete Math.
		On the proof of Andrews' q-Dyson conjecture, pp.
		A combinatorial proof of Schur's 1926 partition theorem.
	www.macalester.edu /~bressoud/pub/biblio.html (281 words)

	[No title]
		One is a combinatorial proof that interprets the LHS and RHS as two different ways of counting the same thing, namely, all the subsets of an n-element set.
		Proof by induction: Left to you to fill in the details.
		Combinatorial proof: We can write the alternating sum sum_k C(n,k) (-1)^k as sum_{S} (-1)^S, where S ranges over all the subsets of [n].
	www.math.wisc.edu /~propp/475/Feb21.doc (301 words)

	A combinatorial proof of Kneser's conjecture - Matousek (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		A combinatorial proof of Kneser's conjecture - Matousek (ResearchIndex)
		By specializing a proof of Tucker's lemma, we obtain self-contained purely combinatorial proof of Kneser's conjecture.
		Matousek, A combinatorial proof of Kneser's conjecture, Combinatorica (to appear).
	citeseer.ist.psu.edu /340774.html (432 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-31)
		Dr. Anthony gave a nice algebraic proof of the equality, but you are right that it wasn't a combinatorial proof.
		Combinatorial proofs rely on counting arguments, not algebraic manipulation.
		When I am working on combinatorial proofs, I find it easier to have just one thing on one of the sides of the equation, because that way it is easier to put into words what a given quantity represents.
	www.mathforum.org /dr.math/problems/turner.7.16.96.html (1069 words)

	Combinatorial proof (Site not responding. Last check: 2007-10-31)
		A combinatorial proof is a method of proving a statement, usually a combinatorial identity, by counting some carefully chosen object in different ways to obtain different expressions in thestatement (see also double counting).
		A statement is said to be proven combinatorially if a combinatorial argument, orcounting argument, is used in the aforementioned fashion to justify the key steps of its proof.
		As the complexity of the problemincreases, a combinatorial proof can become very sophisticated.
	www.therfcc.org /combinatorial-proof-75810.html (380 words)

	Combinatorial proof of the log-concavity of the sequence of matching numbers (Site not responding. Last check: 2007-10-31)
		Combinatorial proof of the log-concavity of the sequence of matching numbers
		For k>=l we construct an injection from the set of pairs of matchings in a given graph G of sizes l-1 and k+1 into the set of pairs of matchings in G of sizes l and k.
		This provides a combinatorial proof of the log-concavity of the sequence of matching numbers of a graph.
	www.mat.univie.ac.at /~kratt/artikel/matching.html (140 words)

	Euler's Formula
		This page lists proofs of the Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.
		Several of the proofs rely on the Jordan curve theorem, which itself has multiple proofs; however these are not generally based on Euler's formula so one can use Jordan curves without fear of circular reasoning.
		Perhaps there is a proof of Euler's formula that uses these polynomials directly rather than merely translating one of the inductions into polynomial form.
	www.ics.uci.edu /%7Eeppstein/junkyard/euler (615 words)

	Notes by David Callan (Site not responding. Last check: 2007-10-31)
		A recent Monthly Note by Oleg Pikhurko gives an "algorithmic" proof that there are n^(n-3) edge-labeled trees on n vertices, reminiscent of Prüfer's proof of Cayley's famous n^(n-2) formula for the number of vertex-labeled trees on n vertices.
		We review combinatorial proofs of (a) and present a new one for (b).
		This note presents an elementary approach to a known combinatorial proof of the cyclotomic identity.
	www.stat.wisc.edu /~callan/papersother (498 words)

	Citebase - A combinatorial proof of the Rogers-Ramanujan and Schur identities (Site not responding. Last check: 2007-10-31)
		Citebase - A combinatorial proof of the Rogers-Ramanujan and Schur identities
		A combinatorial proof of the Rogers-Ramanujan and Schur identities
		We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0411072 (113 words)

	Theory of Computation at Harvard (Site not responding. Last check: 2007-10-31)
		PCP Testers: Towards a combinatorial proof of the PCP Theorem.
		In this work we look back into the proof of the PCP theorem, with the goal of finding new proofs that are either simpler or "more combinatorial".
		While this implies an arguably simpler proof of the PCP theorem, we still rely (in the construction of that basic PCP) on some of the algebraic components of the original proof (most notably, on low-degree tests).
	www.eecs.harvard.edu /theory/03-04/dinurabs.html (234 words)

	discrete mathematics: 1 enumerative volume combinatorics: (Site not responding. Last check: 2007-10-31)
		The material is highbrow (I agree on the 'hardcore' math part, as some people say) but the main theme of the book is on how to 'count' -- needless to say not in the sense of everyday counting, but in the sense that 'topology' is 'coffee-donut transformation' and 'analysis' is 'honors calculus'.
		You have to be able to count combinatorially, and comfortable with combinatorial proof to actually learn from this.
		I like the fact that Stanley asks for combinatorial proof to a some given problems, marking them as unsolved -- he really elevates the status of combinatorial proof, a proof method other mathematicians might dismiss as 'handwaving'.
	www.programming123.com /detail/discrete_mathematics/discrete_mathematics_0521663512.html (385 words)

	mp_arc 03-47 (Site not responding. Last check: 2007-10-31)
		A combinatorial proof of tree decay of semi-invariants (325K, Postscript) Feb 11, 03
		We consider finite range Gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi--invariants, supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions.
		However the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called Griffiths' phase when analyticity arguments fail.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=03-47 (104 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-31)
		Date: 7/17/96 at 8:8:22 From: Doctor Anthony Subject: Re: Combinatorial Proofs This can be proved very quickly if you use the formula for nCr.
		To finish up the proof, we must figure out how to account for this "overcounting." Okay, so to be honest, if we believe that the equation is true, we only have to explain why dividing by C(n-k,r-k) takes care of the overcounting.
		Essentially, combinatorial proofs boil down to providing two different ways to count something.
	mathforum.org /library/drmath/view/56142.html (1069 words)

	University of Minnesota Combinatorics Seminar
		Abstract: We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions.
		This gives a new proof of Tutte's theorem that the number of spanning trees of a central reflex is a perfect square and solves a problem posed by Kalai about higher dimensional spanning trees in simplicial complexes.
		This proof introduces a new matroid invariant which, although it can be described combinatorially, naturally arises from K-theory and has no good combinatorial description as yet.
	www.math.umn.edu /~stanton/seminar.html (2111 words)

	Ivars Peterson's MathTrek - Counting on Fibonacci
		It's possible to interpret Fibonacci numbers in combinatorial terms; in effect, by counting certain tilings.
		It turns out that a Fibonnaci number corresponds to the number of different ways in which you can tile a board of a given length and unit width with squares and dominoes.
		In the book Proofs That Really Count, Arthur T. Benjamin of Harvey Mudd College and Jennifer J. Quinn of Occidental College use this link between Fibonacci numbers and tilings to provide ingenious combinatorial proofs of a wide variety of relationships involving Fibonacci numbers.
	www.maa.org /mathland/mathtrek_05_02_04.html (488 words)

	Stanford Theory Lunch
		This paper presents an abstract mathematical formulation of propositional logic in which proofs are combinatorial (graph-theoretic), rather than syntactic.
		The paper defines a combinatorial proof of a proposition A as a graph homomorphism h : G -> G(A), where G(A) is a graph associated with A, and G is a colored graph.
		Combinatorial proofs are polynomial-time checkable, and provide a possible avenue towards NP=?=coNP.
	theory.stanford.edu /~mihaela/theorylunch (966 words)

	Dan Bernstein
		• A combinatorial proof of a bibasic trigonometric identity
		A combinatorial proof of a bibasic trigonometric identity
		A combinatorial proof of the identity is desired.
	www.wisdom.weizmann.ac.il /~danber (509 words)

	VCP - Visualizing Combinatorial Proofs
		You may find that the background of the proof is sometimes just as fascinating as the proof itself.
		It is important that you actively read the proof (as opposed to speed-reading, skimming, or skipping the proof in favor of the visualization).
		Unlike many proofs presented in textbooks, the combinatorial proofs within these expositions include a lot of details and intermediate steps.
	www.mathcs.bethel.edu /~gossett/vcp (1116 words)

	ECCC Report TR05-046 and related Papers (Site not responding. Last check: 2007-10-31)
		The {em satisfiability-gap} of C is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the variables.
		We prove a new combinatorial amplification lemma that doubles the satisfiability-gap of a constraint-system, with only a linear blowup in the size of the system.
		Iterative application of this lemma yields a self-contained (combinatorial) proof for the PCP theorem.
	eccc.hpi-web.de /eccc-reports/2005/TR05-046/index.html (225 words)

	Geometry.Net - Pure And Applied Math Books: Combinatorics
		It is the the advent of the computer though that has had the greatest influence on combinatorics, and vice versa.The consideration of NP complete problems typically involves enumerative problems in graph theory, one example being the existance of a Hamiltonian cycle in a graph.
		The use of the computer as a tool for proof in combinatorics, such as the 4-color problem, is now legendary.
		Combinatorial techniques had a large role to play in the problem of the classification of finite simple groups, the eventual classification proof taking over 15,000 journal pages and involving a large collaboration of mathematicians.
	www.geometry.net /pure_and_applied_math_bk/combinatorics_page_no_2.html (3459 words)

	[No title]
		Abstract: We provide the first combinatorial proof for the sum of the cubes of the first n Fibonacci numbers.
		Abstract:We give a combinatorial proof of the first Rogers-Ramanujan identity by definining a new generalization of Dyson's rank and presenting two related symmetries.
		Abstract: We discuss a recent result of M. Haiman, N. Loehr, and the speaker, which gives a combinatorial formula, involving generalizations of the permutation statistics maj and inv, for the coefficient of a monomial in the modified Macdonald polynomial.
	www.math.rutgers.edu /~asills/expmath/fall04.html (949 words)

	Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions) (Site not responding. Last check: 2007-10-31)
		While the theorems covered may not be new to research mathematicians, I would wager that very few of us have seen them proven in quite this way." -- American Mathematical Monthly [http://www.maa.org/reviews/reallycount.html] lt;br /gt; lt;br /gt;I am not a mathematician and I learn something cool and useful from this book every few paragraphs.
		I was introduced to this book by a talk that one of the authors (Arthur Benjamin) gave at the MAA Mathfest in Albuquerque in August of 2005.
		Everyone in the audience could follow what was going on, and we all left with an understanding of the basic approach to combinatorial identities used in this book.
	www.history-us.com /Proofs_that_Really_Count__The_Art_of_Combinatorial_Proof_0883853337.html (407 words)

	Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions) (Site not responding. Last check: 2007-10-31)
		The authors approach is to prove combinatorial identities by defining a quantity and then obtaining different formulas for that quantity.
		If her written work is anything like her speaking, then this should be a great book.
		Her combinatorial proofs are an interesting approach to old equations, and she presents them in a very clear manner.
	www.psych-books.com /Proofs_that_Really_Count__The_Art_of_Combinatorial_Proof_Dolciani_Mathematical_Expositions_0883853337.html (407 words)

	Lynne Butler's Undergraduate Research Projects
		For type (1,1,...,1), this result is the "p-analogue" of the fact that the number of subsets of cardinality k equals the number of subsets of cardinality n-k in a set of cardinality n.
		A combinatorial proof of this fact pairs each subset with its complement.
		Michael Aguilar Combinatorial properties of the lattice of subgroups of a finite abelian p-group, senior thesis, Princeton University, 1990.
	www.haverford.edu /math/lbutler/research.html (1223 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		TITLE: PCP Testers: Towards a combinatorial proof of the PCP Theorem ABSTRACT: In this work we look back into the proof of the PCP theorem, with the goal of finding new proofs that are ``more combinatorial" and arguably simpler.
		This proof relies on a rather weak PCP given as a ``fl box".
		From this, we construct combinatorially the full PCP, relying on composition and a new combinatorial aggregation technique.
	www.cs.huji.ac.il /~theorys/abstracts/Irit_dinur.txt (161 words)

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