Vandermonde's identity: Difference between revisions

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In [[combinatorics]], '''bijective proof''' is a [[mathematical proof|proof]] technique that finds a [[bijective function]] ''f'' : ''A'' → ''B'' between two [[finite set]]s ''A'' and ''B'', or a size-preserving bijective function between two [[combinatorial class]]es, thus proving that they have the same number of elements, |''A''| = |''B''|. One place the technique is useful is where we wish to know the size of ''A'', but can find no direct way of counting its elements.  Then establishing a bijection from ''A'' to some ''B'' solves the problem in the case when ''B'' is more easily countable.  Another useful feature of the technique is that the nature of the bijection itself often provides powerful insights into each or both of the sets.
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=== Proving the symmetry of the binomial coefficients ===
  <li>[http://www.dpshunyi.com/bbs/forum.php?mod=viewthread&tid=2088896&fromuid=174036 http://www.dpshunyi.com/bbs/forum.php?mod=viewthread&tid=2088896&fromuid=174036]</li>
 
 
The symmetry of the binomial coefficients states that
  <li>[http://www.cpefound.org.php5-14.dfw1-1.websitetestlink.com/node/13#comment-92473 http://www.cpefound.org.php5-14.dfw1-1.websitetestlink.com/node/13#comment-92473]</li>
 
 
:<math> {n \choose k} = {n \choose n-k}. </math>
  <li>[http://09dm.com/news/html/?180898.html http://09dm.com/news/html/?180898.html]</li>
 
 
This means there are exactly as many [[permutations and combinations|combinations]] of ''k'' in a set of ''n'' as there are combinations of ''n''&nbsp;&minus;&nbsp;''k'' in a set of&nbsp;''n''.
  <li>[http://c987547.s08.108198.com/forum.php?mod=viewthread&tid=689272&fromuid=51760 http://c987547.s08.108198.com/forum.php?mod=viewthread&tid=689272&fromuid=51760]</li>
 
 
==== The bijective proof ====
  <li>[http://fuxiaoliming.com/forum.php?mod=viewthread&tid=228521&extra= http://fuxiaoliming.com/forum.php?mod=viewthread&tid=228521&extra=]</li>
 
 
More abstractly and generally, we note that the two quantities asserted to be equal count the subsets of size ''k'' and ''n''&nbsp;&minus;&nbsp;''k'', respectively, of any ''n''-element set ''S''. There is a simple bijection between the two families ''F''<sub>''k''</sub> and ''F''<sub>''n''&nbsp;&minus;&nbsp;''k''</sub> of subsets of ''S'': it associates every ''k''-element subset with its [[complement (set theory)|complement]], which contains precisely the remaining ''n''&nbsp;&minus;&nbsp;''k'' elements of ''S''. Since ''F''<sub>''k''</sub> and ''F''<sub>''n''&nbsp;&minus;&nbsp;''k''</sub> have the same number of elements, the corresponding binomial coefficients must be equal.
</ul>
 
=== Pascal's triangle recurrence relation ===
:<math> {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}\text{ for }1 \le k \le n - 1. </math>
 
==== The bijective proof ====
 
''Proof''.
We count the number of ways to choose ''k'' elements from an ''n''-set.
Again, by definition, the left hand side of the equation is the number of ways to choose ''k'' from ''n''.
Since 1 ≤ ''k'' ≤ ''n'' &minus; 1, we can pick a fixed element ''e'' from the ''n''-set so that the remaining subset is not empty.
For each ''k''-set, if ''e'' is chosen, there are
 
: <math>{n-1 \choose k-1}</math>
 
ways to choose the remaining ''k''&nbsp;&minus;&nbsp;1 elements among the remaining ''n''&nbsp;&minus;&nbsp;1 choices; otherwise, there are  
 
: <math>{n-1 \choose k}</math>
 
ways to choose the remaining ''k'' elements among the remaining ''n'' &minus; 1 choices. Thus, there are  
 
:<math>{n-1 \choose k-1} + {n-1 \choose k}</math>
 
ways to choose ''k'' elements depending on whether ''e'' is included in each selection, as in the right hand side expression. <math>\Box</math>
 
== Other examples ==
 
Problems that admit combinatorial proofs are not limited to binomial coefficient identities. As the complexity of the problem increases, a combinatorial proof can become very sophisticated. This technique is particularly useful in areas of [[discrete mathematics]] such as [[combinatorics]], [[graph theory]], and [[number theory]].
 
The most classical examples of bijective proofs in combinatorics include:
* [[Prüfer sequence]], giving a proof of [[Cayley's formula]] for the number of [[labeled tree]]s.
* [[Robinson-Schensted algorithm]], giving a proof of [[William Burnside|Burnside]]'s formula for the [[symmetric group]].
* [[Partition (number theory)#Ferrers graph|Conjugation]] of [[Young diagram]]s, giving a proof of a classical result on the number of certain [[Partition (number theory)|integer partitions]].
* Bijective proofs of the [[pentagonal number theorem]].
* Bijective proofs of the formula for the [[Catalan number]]s.
 
== See also==
* [[Binomial theorem]]
* [[Cantor–Bernstein–Schroeder theorem]]
* [[Double counting (proof technique)]]
* [[Combinatorial principles]]
* [[Combinatorial proof]]
* [[Categorification]]
 
== External links ==
*''[http://www.math.dartmouth.edu/~doyle/docs/three/three.pdf "Division by three"]'' – by Doyle and [[John Horton Conway|Conway]].
*''[http://www.emis.ams.org/journals/DMTCS/volumes/abstracts/pdfpapers/dm010104.pdf "A direct bijective proof of the hook-length formula"]'' –  by Novelli, [[Igor Pak|Pak]] and Stoyanovsky.
*''[http://www.emis.ams.org/journals/EJC/Volume_4/PDF/v4i1r20.pdf "Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees"]'' –  by Gilles Schaeffer.
*''[http://www.math.temple.edu/~zeilberg/mamarim/mamarimPDF/ohara.pdf "Kathy O'Hara's Constructive Proof of the Unimodality of the Gaussian Polynomials"]'' – by [[Doron Zeilberger]].
*''[http://www.math.umn.edu/~pak/psurvey.pdf "Partition Bijections, a Survey"]'' – by [[Igor Pak]].
*[http://mathworld.wolfram.com/Garsia-MilneInvolutionPrinciple.html Garsia-Milne Involution Principle] – from [[MathWorld]].
 
==References==
* Loehr, Nicholas A. (2011).  [http://www.math.vt.edu/people/nloehr/bijbook.html Bijective Combinatorics].  [http://www.crcpress.com CRC Press]. ISBN  143984884X,  ISBN  978-1439848845.
 
{{DEFAULTSORT:Bijective Proof}}
[[Category:Enumerative combinatorics]]
[[Category:Articles containing proofs]]
[[Category:Mathematical proofs]]

Latest revision as of 12:16, 8 January 2015

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