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href="/topics/MathematicsintheArts.html">Mathematics in the Arts</a> </li> <li> <a href="/topics/MathematicsinTelevision.html">Mathematics in Television</a> </li> <li> <a href="/topics/NUMB3RS.html">NUMB3RS</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Renze.html">Renze</a> </li> </ul></nav>   <h1>Combinatorics</h1>  <hr class="margin-t-1-8 margin-b-3-4">   <div class="entry-content"> <p> Combinatorics is the branch of mathematics studying the <a href="/Enumeration.html">enumeration</a>, <a href="/Combination.html">combination</a>, and <a href="/Permutation.html">permutation</a> of <a href="/Set.html">sets</a> of elements and the mathematical relations that characterize their properties. </p> <p> Mathematicians sometimes use the term "combinatorics" to refer to a larger subset of <a href="/DiscreteMathematics.html">discrete mathematics</a> that includes <a href="/GraphTheory.html">graph theory</a>. In that case, what is commonly called combinatorics is then referred to as "<a href="/Enumeration.html">enumeration</a>." </p> <p> The Season 1 episode "<a href="http://www.tv.com/numb3rs/episode/398602/summary.html">Noisy Edge</a>" (2005) of the television crime drama <i><a href="http://www.amazon.com/exec/obidos/ASIN/B000ERVJKE/ref=nosim/ericstreasuretro">NUMB3RS</a></i> mentions combinatorics. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/AlgebraicCombinatorics.html">Algebraic Combinatorics</a>, <a href="/Antichain.html">Antichain</a>, <a href="/Chain.html">Chain</a>, <a href="/ConcreteMathematics.html">Concrete Mathematics</a>, <a href="/DilworthsLemma.html">Dilworth's Lemma</a>, <a href="/DirichletsBoxPrinciple.html">Dirichlet's Box Principle</a>, <a href="/DiscreteMathematics.html">Discrete Mathematics</a>, <a href="/EnumerationProblem.html">Enumeration Problem</a>, <a href="/Erdos-SzekeresTheorem.html">Erdős-Szekeres Theorem</a>, <a href="/Inclusion-ExclusionPrinciple.html">Inclusion-Exclusion Principle</a>, <a href="/KirkmansSchoolgirlProblem.html">Kirkman's Schoolgirl Problem</a>, <a href="/KirkmanTripleSystem.html">Kirkman Triple System</a>, <a href="/PartialOrder.html">Partial Order</a>, <a href="/PartialOrderLength.html">Partial Order Length</a>, <a href="/PartialOrderWidth.html">Partial Order Width</a>, <a href="/RamseysTheorem.html">Ramsey's Theorem</a>, <a href="/Schroeder-BernsteinTheorem.html">Schröder-Bernstein Theorem</a>, <a href="/SchursLemma.html">Schur's Lemma</a>, <a href="/SpernersTheorem.html">Sperner's Theorem</a>, <a href="/TotalOrder.html">Total Order</a>, <a href="/UmbralCalculus.html">Umbral Calculus</a>, <a href="/vanderWaerdensTheorem.html">van der Waerden's Theorem</a> <a href="/classroom/Combinatorics.html" class="explore-classroom">Explore this topic in the MathWorld classroom</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=combinatorics"> combinatorics </a> </li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=7.5%25+of+%2412.95">7.5% of .95</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=d%2Fdx+x%5E2+y%5E4%2C+d%2Fdy+x%5E2+y%5E4">d/dx x^2 y^4, d/dy x^2 y^4</a></li> </ul> </div> </div>   <h2>References</h2><cite>Abramowitz, M. and Stegun, I. A. (Eds.). "Combinatorial Analysis." Ch. 24 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486612724/ref=nosim/ericstreasuretro">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.</a></i> New York: Dover, pp. 821-827, 1972.</cite><cite>Aigner, M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/3540617876/ref=nosim/ericstreasuretro">Combinatorial Theory.</a></i> New York: Springer-Verlag, 1997.</cite><cite>Balakrishnan, V. K. <i><a href="http://www.amazon.com/exec/obidos/ASIN/007003575X/ref=nosim/ericstreasuretro">Schaum's Outline of Combinatorics, including Concepts of Graph Theory.</a></i> New York: McGraw-Hill, 1995.</cite><cite>Bellman, R. and Hall, M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821813102/ref=nosim/ericstreasuretro">Combinatorial Analysis.</a></i> Amer. Math. Soc., 1979.</cite><cite>Berge, C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0120897504/ref=nosim/ericstreasuretro">Principles of Combinatorics.</a></i> New York: Academic Press, 1971.</cite><cite>Bergeron, F.; Labelle, G.; and Leroux, P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521573238/ref=nosim/ericstreasuretro">Combinatorial Species and Tree-Like Structures.</a></i> Cambridge, England: Cambridge University Press, 1998.</cite><cite>Biggs, N. L. "The Roots of Combinatorics." <i>Historia Mathematica</i> <b>6</b>, 109-136, 1979.</cite><cite>Bose, R. C. and Manvel, B. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471896144/ref=nosim/ericstreasuretro">Introduction to Combinatorial Theory.</a></i> New York: Wiley, 1984.</cite><cite>Cameron, P. J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521457610/ref=nosim/ericstreasuretro">Combinatorics: Topics, Techniques, Algorithms.</a></i> New York: Cambridge University Press, 1994.</cite><cite>Cohen, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471035351/ref=nosim/ericstreasuretro">Basic Techniques of Combinatorial Theory.</a></i> New York: Wiley, 1978.</cite><cite>Cohen, D. E. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521349362/ref=nosim/ericstreasuretro">Combinatorial Group Theory: A Topological Approach.</a></i> New York: Cambridge University Press, 1989.</cite><cite>Colbourn, C. J. and Dinitz, J. H. (Eds.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0849389488/ref=nosim/ericstreasuretro">CRC Handbook of Combinatorial Designs.</a></i> Boca Raton, FL: CRC Press, 1996.</cite><cite>MathPages. "Combinatorics." <a href="http://www.mathpages.com/home/icombina.htm">http://www.mathpages.com/home/icombina.htm</a>.</cite><cite>Comtet, L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/9027703809/ref=nosim/ericstreasuretro">Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed.</a></i> Dordrecht, Netherlands: Reidel, 1974.</cite><cite><a href="/contribute/updated_hyperlink.html#Combinatorics"><img src="/images/entries/contribute_link.gif" width="88" height="24" border="0" alt="Update a link" /></a>Coolsaet, K. "Index of Combinatorial Objects." <a href="http://gonzo.hogent.be/~kc/ico/">http://gonzo.hogent.be/~kc/ico/</a></cite><cite>Dinitz, J. H. and Stinson, D.  R. (Eds.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471531413/ref=nosim/ericstreasuretro">Contemporary Design Theory: A Collection of Surveys.</a></i> New York: Wiley, 1992.</cite><cite>Eisen, M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0677022603/ref=nosim/ericstreasuretro">Elementary Combinatorial Analysis.</a></i> New York: Gordon and Breach, 1969.</cite><cite><i>Electronic Journal of Combinatorics.</i> <a href="http://www.combinatorics.org/previous_volumes.html">http://www.combinatorics.org/previous_volumes.html</a>.</cite><cite>Eppstein, D. "Combinatorial Geometry." <a href="http://www.ics.uci.edu/~eppstein/junkyard/combinatorial.html">http://www.ics.uci.edu/~eppstein/junkyard/combinatorial.html</a>.</cite><cite>Erdős, P. and Spencer, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0122409604/ref=nosim/ericstreasuretro">Probabilistic Methods in Combinatorics.</a></i> New York: Academic Press, 1974.</cite><cite>Erickson, M. J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471154083/ref=nosim/ericstreasuretro">Introduction to Combinatorics.</a></i> New York: Wiley, 1996.</cite><cite>Fields, J. "On-Line Dictionary of Combinatorics." <a href="http://www.math.uic.edu/~fields/comb_dic/">http://www.math.uic.edu/~fields/comb_dic/</a>.</cite><cite>Gardner, M. "Combinatorial Theory." Ch. 3 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0226282503/ref=nosim/ericstreasuretro">The Sixth Book of Mathematical Games from Scientific American.</a></i> Chicago, IL: University of Chicago Press, pp. 19-28, 1984.</cite><cite>Godsil, C. D. "Problems in Algebraic Combinatorics." <i>Electronic J. Combinatorics</i> <b>2</b>, No. 1, R1, 1-20, 1995. <a href="http://www.combinatorics.org/Volume_2/Abstracts/v2i1r1.html">http://www.combinatorics.org/Volume_2/Abstracts/v2i1r1.html</a>.</cite><cite>Graham, R. L.; Grötschel, M.; and Lovász, L. (Eds.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0262571722/ref=nosim/ericstreasuretro">Handbook of Combinatorics, 2 vols.</a></i> Cambridge, MA: MIT Press, 1996.</cite><cite>Graham, R. L.; Knuth, D. E.; and Patashnik, O. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201558025/ref=nosim/ericstreasuretro">Concrete Mathematics: A Foundation for Computer Science, 2nd ed.</a></i> Reading, MA: Addison-Wesley, 1994.</cite><cite>Grimaldi, R. P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201199122/ref=nosim/ericstreasuretro">Discrete and Combinatorial Mathematics: An Applied Introduction, 4th ed.</a></i> Longman, 1998.</cite><cite>Hall, M. Jr. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471315184/ref=nosim/ericstreasuretro">Combinatorial Theory, 2nd ed.</a></i> New York: Wiley, 1986.</cite><cite>Harary, F. <i>Applied Combinatorial Mathematics.</i> New York: Wiley, 1964.</cite><cite>Knuth, D. E. (Ed.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821806033/ref=nosim/ericstreasuretro">Stable Marriage and Its Relation to Other Combinatorial Problems.</a></i> Providence, RI: Amer. Math. Soc., 1997.</cite><cite>Kreher, D. L. and Stinson, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/084933988X/ref=nosim/ericstreasuretro">Combinatorial Algorithms: Generation, Enumeration, and Search.</a></i> Boca Raton, FL: CRC Press, 1999.</cite><cite>Kučera, L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0852742983/ref=nosim/ericstreasuretro">Combinatorial Algorithms.</a></i> Bristol, England: Adam Hilger, 1989.</cite><cite>Liu, C. L. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0070381240/ref=nosim/ericstreasuretro">Introduction to Combinatorial Mathematics.</a></i> New York: McGraw-Hill, 1968.</cite><cite>MacMahon, P. A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0828411379/ref=nosim/ericstreasuretro">Combinatory Analysis, 2 vols.</a></i> New York: Chelsea, 1960.</cite><cite>Marcus, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0883857081/ref=nosim/ericstreasuretro">Combinatorics: A Problem Oriented Approach.</a></i> Washington, DC: Math. Assoc. Amer., 1998.</cite><cite>Nijenhuis, A. and Wilf, H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0125192606/ref=nosim/ericstreasuretro">Combinatorial Algorithms for Computers and Calculators, 2nd ed.</a></i> New York: Academic Press, 1978.</cite><cite>Petit, S. "Encyclopedia of Combinatorial Structures." <a href="http://algo.inria.fr/encyclopedia/">http://algo.inria.fr/encyclopedia/</a>.</cite><cite>Raghavarao, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486656853/ref=nosim/ericstreasuretro">Constructions and Combinatorial Problems in Design of Experiments.</a></i> New York: Dover, 1988.</cite><cite>Riordan, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0882758292/ref=nosim/ericstreasuretro">Combinatorial Identities, reprint ed. with corrections.</a></i> Huntington, NY: Krieger, 1979.</cite><cite>Riordan, J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0691023654/ref=nosim/ericstreasuretro">An Introduction to Combinatorial Analysis.</a></i> New York: Wiley, 1980.</cite><cite>Roberts, F. S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0130393134/ref=nosim/ericstreasuretro">Applied Combinatorics.</a></i> Englewood Cliffs, NJ: Prentice-Hall, 1984.</cite><cite>Rosen, K. H. (Ed.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0849301491/ref=nosim/ericstreasuretro">Handbook of Discrete and Combinatorial Mathematics.</a></i> Boca Raton, FL: CRC Press, 2000.</cite><cite>Rota, G.-C. (Ed.). <i><a href="http://www.amazon.com/exec/obidos/ASIN/0883851172/ref=nosim/ericstreasuretro">Studies in Combinatorics.</a></i> Providence, RI: Math. Assoc. Amer., 1978.</cite><cite>Ruskey, F. "The (Combinatorial) Object Server." <a href="http://www.theory.csc.uvic.ca/~cos/">http://www.theory.csc.uvic.ca/~cos/</a>.</cite><cite>Ryser, H. J. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0883850141/ref=nosim/ericstreasuretro">Combinatorial Mathematics.</a></i> Buffalo, NY: Math. Assoc. Amer., 1963.</cite><cite>Skiena, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521806860/ref=nosim/ericstreasuretro">Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.</a></i> Reading, MA: Addison-Wesley, 1990.</cite><cite>Sloane, N. J. A. "An On-Line Version of the Encyclopedia of Integer Sequences." <a href="http://www.research.att.com/~njas/sequences/eisonline.html">http://www.research.att.com/~njas/sequences/eisonline.html</a>.</cite><cite>Sloane, N. J. A. and Plouffe, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0125586302/ref=nosim/ericstreasuretro">The Encyclopedia of Integer Sequences.</a></i> San Diego, CA: Academic Press, 1995.</cite><cite>Slomson, A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0412353709/ref=nosim/ericstreasuretro">Introduction to Combinatorics.</a></i> Boca Raton, FL: Chapman and Hall, 1997.</cite><cite>Stanley, R. P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521663512/ref=nosim/ericstreasuretro">Enumerative Combinatorics, Vol. 1.</a></i> Cambridge, England: Cambridge University Press, 1999.</cite><cite>Stanley, R. P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521560691/ref=nosim/ericstreasuretro">Enumerative Combinatorics, Vol. 2.</a></i> Cambridge, England: Cambridge University Press, 1999.</cite><cite>Street, A. P. and Wallis, W. D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/B0006CW9V0/ref=nosim/ericstreasuretro">Combinatorial Theory: An Introduction.</a></i> Winnipeg, Manitoba: Charles Babbage Research Center, 1977.</cite><cite>Tucker, A. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0471595047/ref=nosim/ericstreasuretro">Applied Combinatorics, 3rd ed.</a></i> New York: Wiley, 1995.</cite><cite>van Lint, J. H. and Wilson, R. M. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521422604/ref=nosim/ericstreasuretro">A Course in Combinatorics.</a></i> New York: Cambridge University Press, 1992.</cite><cite>Weisstein, E. W. "Books about Combinatorics." <a href="http://www.ericweisstein.com/encyclopedias/books/Combinatorics.html">http://www.ericweisstein.com/encyclopedias/books/Combinatorics.html</a>.</cite><cite>Wilf, H. S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0898712319/ref=nosim/ericstreasuretro">Combinatorial Algorithms: An Update.</a></i> Philadelphia, PA: SIAM, 1989.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/combinatorics/u1/u5/h3/" title="Combinatorics" target="_blank">Combinatorics</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Combinatorics." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/Combinatorics.html">https://mathworld.wolfram.com/Combinatorics.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/Combinatorics.html">Combinatorics</a> </li> <li> <a href="/topics/GeneralCombinatorics.html">General Combinatorics</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/RecreationalMathematics.html">Recreational Mathematics</a> </li> <li> <a href="/topics/MathematicsintheArts.html">Mathematics in the Arts</a> </li> <li> <a href="/topics/MathematicsinTelevision.html">Mathematics in Television</a> </li> <li> <a href="/topics/NUMB3RS.html">NUMB3RS</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Renze.html">Renze</a> </li> </ul></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,208 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Thu Nov 21 2024</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2024 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>