• the term combinatorial proof is often used to mean either of two types of mathematical proof: A proof by double counting. A combinatorial identity is...
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    Benjamin, Arthur T.; Quinn, Jennifer J. (2003). Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Vol. 27. Mathematical...
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  • a simplified combinatorial proof formulated in the terminology of robot arm motion planning. Both the original proof and Streinu's proof work by finding...
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  • come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately...
    61 KB (8,516 words) - 04:02, 21 May 2025
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    for testing primality) are as good as genuine mathematical proofs. A combinatorial proof establishes the equivalence of different expressions by showing...
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    ; Todd, Michael J. (1982). "A constructive proof of Tucker's combinatorial lemma". Journal of Combinatorial Theory. Series A. 30 (3): 321–325. doi:10...
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    Cayley–Hamilton theorem (category Articles containing proofs)
    1997, p. 7 Garrett 2007, p. 381 Straubing, Howard (1983-01-01). "A combinatorial proof of the Cayley-Hamilton theorem". Discrete Mathematics. 43 (2): 273–279...
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  • combinatorics, bijective proof is a proof technique for proving that two sets have equally many elements, or that the sets in two combinatorial classes have equal...
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  • the simplest known proof, requiring the least mathematical background. It is an attractive example of a combinatorial proof (a proof that involves counting...
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  • the formalisation of intuitionistic logic, and provide the first combinatorial proof of the consistency of Peano arithmetic. Together, the presentation...
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  • Pascal's rule has an intuitive combinatorial meaning, that is clearly expressed in this counting proof.: 44  Proof. Recall that ( n k ) {\displaystyle...
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  • Binomial theorem (category Articles containing proofs)
    {\displaystyle {\tbinom {n}{k}},} either by definition, or by a short combinatorial argument if one is defining ( n k ) {\displaystyle {\tbinom {n}{k}}}...
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  • Proofs That Really Count: the Art of Combinatorial Proof is an undergraduate-level mathematics book on combinatorial proofs of mathematical identies....
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  • Vandermonde's identity (category Articles containing proofs)
    binomial coefficients. Vandermonde's identity also admits a combinatorial double counting proof, as follows. Suppose a committee consists of m men and n...
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    Fibonacci sequence (category Articles containing proofs)
    memoization). Most identities involving Fibonacci numbers can be proved using combinatorial arguments using the fact that F n {\displaystyle F_{n}} can be interpreted...
    86 KB (13,070 words) - 08:03, 31 May 2025
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    Hockey-stick identity (category Articles containing proofs)
    {\text{ for }}n,r\in \mathbb {N} ,\quad n\geq r.} The inductive and algebraic proofs both make use of Pascal's identity: ( n k ) = ( n − 1 k − 1 ) + ( n − 1...
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  • MathWorld A Matrix Proof of Newton's Identities in Mathematics Magazine Application on the number of real roots A Combinatorial Proof of Newton's Identities...
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  • In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate...
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  • Binomial coefficients and their properties Combinatorial proof Double counting (proof technique) Bijective proof Inclusion–exclusion principle Möbius inversion...
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  • combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating...
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    required a computer-assisted proof. Many real-world games remain too complex for complete analysis, though combinatorial methods have shown some success...
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  • now called Polymath1 by the Polymath community, was to find a new combinatorial proof to the density version of the Hales–Jewett theorem. As the project...
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  • {\displaystyle |Y|\setminus |X|} . Björner and Tancer presented an elementary combinatorial proof and summarized a few generalizations. For smooth manifolds, Alexander...
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    Press. ISBN 978-0-88385-700-7. Stein, Robert G. (1 May 1971). "A Combinatorial Proof That Σ k3 = (Σ k)2". Mathematics Magazine. 44 (3): 161–162. doi:10...
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  • original on 2016-08-26. Retrieved 2007-11-25. Fitzi, Matthias. "Combinatorial Optimization via Proof-of-Useful-Work" (PDF). IACR conference Crypto 2022. Archived...
    37 KB (4,073 words) - 20:11, 27 May 2025
  • called "k-permutations" of the n-set.) Straubing, Howard (1983), "A combinatorial proof of the Cayley-Hamilton theorem", Discrete Mathematics, 43 (2–3):...
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  • "Editorial". Combinatorial Theory. 1. doi:10.5070/C61055307. ISSN 2766-1334. S2CID 245076810. Katona, G.O.H. (1972). "A simple proof of the Erdös-Chao...
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  • Counting argument may refer to: Pigeonhole principle Combinatorial proof This disambiguation page lists articles associated with the title Counting argument...
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  • Jennifer J. (2003). "Fibonacci and Lucas Polynomial". Proofs that Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Vol. 27. Mathematical...
    8 KB (1,612 words) - 07:23, 28 May 2024
  • In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule...
    5 KB (727 words) - 16:43, 10 February 2024