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Writing a recursive definition of the set permutation function

  1. Feb 2, 2012 #1
    (Apologies for the lack of LaTeX formatting - I usually do my typesetting with MathType, but I only have Microsoft Equation Editor on this computer which doesn't have LaTeX export).

    This isn't a homework question, but is problem I've discovered I'm facing after diving into another problem, but I digress:

    Consider this set, A:

    A = { T, T, R }

    It has six proper permutations, but only three distinct permutations:

    TTR, TRT, RTT

    The expression for the number of unique permutations (in this case) is:

    |A|! / ( |'T' elements| ! * |'R' elements| ! )

    So: 3! / 2! * 1! which is 6/2 == 3

    Due to certain requirements, I need to express this function recursively. Because the factorial function is used, I figured I could take it from there.

    The expressions below take the permutation-count formula and assumes that every set A is comprised of N 'T' elements, and M 'R' elements and is called "routes".

    KSsTC.png

    ...but I can't figure out how to make it recursive. Note the huge error on the third line down, it does not follow from the line above.

    Is a recursive definition even possible?

    Thanks!
     
  2. jcsd
  3. Feb 3, 2012 #2
    You can express a binomial coefficient recursively like this (see wikipedia article):
    [tex]\begin{pmatrix} n \\ k \end{pmatrix} = \begin{pmatrix} n-1 \\ k-1 \end{pmatrix} + \begin{pmatrix} n-1 \\ k \end{pmatrix}[/tex]

    You can interpret this as Pascal's triangle.
     
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