Combinatorics (Fall 2010)/Basic enumeration

The twelvfold way

$f:N\to M$

Elements of $N$	Elements of $M$	Any $f$	Injective (1-1) $f$	Surjective (on-to) $f$
distinguishable	distinguishable	$m^n$	${\left(m\right)}_n$	$m!S(n,m)$
indistinguishable	distinguishable	$\left((\genfrac{}{}{0ex}{}{m}{n})\right)$	$(\genfrac{}{}{0ex}{}{m}{n})$	$\left((\genfrac{}{}{0ex}{}{m}{n-m})\right)$
distinguishable	indistinguishable	$\sum _{k=1}^{m}S(n,k)$	$\{\begin{array}{ll}1& \text{if }n\le m\\ 0& \text{if }n>m\end{array}$	$S(n,m)$
indistinguishable	indistinguishable	$\sum _{k=1}^m{p}_k(n)$	$\{\begin{array}{ll}1& \text{if }n\le m\\ 0& \text{if }n>m\end{array}$	$p_m(n)$