The twelvefold way gives the number of mappings 
from set 
of 
objects to set 
of 
objects (putting balls from set into boxes in set
). See Table 26.17.1. In this table 
is
Pochhammer’s symbol, and 
and 
are
defined in §§26.8(i) and 26.9(i).
Table 26.17.1 is reproduced (in modified form) from Stanley (1997, p. 33). See also Example 3 in §26.18.
| elements of
|
elements of
|
unrestricted |
one-to-one |
onto |
|---|---|---|---|---|
| labeled | labeled |  |
 |
 |
| unlabeled | labeled |  |
 |
 |
| labeled | unlabeled | ![\begin{array}[]{c}\mathop{S\/}\nolimits\!\left(n,1\right)+\mathop{S\/}%
\nolimits\!\left(n,2\right)\\
+\cdots+\mathop{S\/}\nolimits\!\left(n,k\right)\end{array}](./26/17/m7.png){kind=small} |
 |
|
| unlabeled | unlabeled | |
|
 |
: Stirling number of the second kind,
: binomial coefficient,
: 
: factorial,
: total number of partitions of ,
: nonnegative integer,
: nonnegative integer,
: mapping and
: set
