Introduction E9F5FC

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Reference

• Notes on the combinatorial fundamentals of algebra. Darij Grinberg.

{$$\binom{X}{n} = \frac{X(X-1)\cdots(X-m-1)}{m!}$$}

This gives the values down the th diagonal, counting from 0:

{$$\binom{X}{0} = 1$$}

{$$\binom{X}{1} = X$$}

{$$\binom{X}{2} = \frac{X^2-X}{2}$$}

{$$\binom{X}{3} = \frac{X^3-3X^2+2X}{6}$$}

The recursion is given by

{$$\binom{X}{m}=\binom{X-1}{m-1}+\binom{X-1}{m}$$}

{$$\frac{X(X-1)\cdots(X-m+1)}{m!} = \frac{(X-1)\cdots(X-m+1)}{m-1!} + \frac{(X-1)\cdots(X-m)}{m!}$$}

{$$ = \frac{(X-1)\cdots(X-m+1)}{m-1!}(1 + \frac{X-m}{m})$$}

{$$ = \frac{(X-1)\cdots(X-m+1)}{m-1!}(\frac{m+X-m}{m})$$}

{$$ = \frac{(X-1)\cdots(X-m)}{m-1!}(\frac{X}{m})$$}

What is the combinatorial interpretation?