Revistes Catalanes amb Accés Obert (RACO)

Application of Cauchy's equation in combinatorics and genetics

Palaniappan Kannappan

Resum


We are familiar with the
combinatorial formula
$\left(\begin{array}{cc}
n\\
r
\end{array}\right) = \frac{n(n-1) \cdots (n - r +
1)}{r !} = $ number
of possible ways of choosing $r$ objects out of $n$ objects\,.


In section 1 of this paper we obtain $\left(
\begin{array}{cc} n\\
2\end{array}\right)$ and $\left( \begin{array}{cc}
n\\
3
\end{array}\right)$ by using a functional equation, {\it
the additive Cauchy equation}.

In genetics it is important to know the combinatorial
function $g_{r}(n)=$ the number of possible ways of
picking $r$ objects at a time from $n$ objects {\it
allowing repetitions}, since this function describes the
number of possibilities from a gene pool. Again we determine $g_2(n)$ and $g_3(n)$ with the help of the additive Cauchy equation in section 2.

Functional equations are used increasingly in diverse
fields. The method of finding $\left( \begin{array}{cc}
n\\
2
\end{array}\right), \left( \begin{array}{cc}
n\\
3
\end{array}\right), g_2 (n)$ and $g_3(n)$ (see Snow [6])
is similar to that of finding the well known sum of
powers of integers $S_K(n) = 1^K + 2^K + \cdots + n^K$
(Acz\'{e}l [2], Snow [5]).\\

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