Counting Table

Cells $n$	distinct	distinct	identical	identical
Balls $k$	distinct	identical	distinct	identical
$0 \leq x_{i} \leq 1$ , $n \geq k$	$\frac{n !}{( n - k )!} = (n)_{k}$	$(k n)$	1	1
$x_{i} \geq 0$	$n^{k}$	$(k n + k - 1)$		$p (k)$ ^[Partition]
$x_{i} \geq 1$	$n! {n k}$	$(n - 1 k - 1)$		$p_{n} (k)$ ^[ [[Partition#Partition into k]]. Note, there, n is balls, and k is cells ]

Shahriari, An Invitation to Combinatorics, 349.

#	$k$ Balls	$n$ Cells	Order in Cell	Empty Cells	Additional Restrictions	Formula / Notation
1	Identical	Distinct	No	allowed	At most 1 per cell	$(k n)$ if $k \leq n$
2	Distinct	Distinct	No	allowed	At most 1 per cell	$(k n) \cdot k!$ if $k \leq n$
3	Distinct	Distinct	No	allowed	None	$n^{k}$
4	Identical	Distinct	No	allowed	None	$(k n + k - 1)$
5	Distinct	Distinct	Yes	allowed	None	$n^{k}$
6	Distinct	Distinct	No	Not allowed	None	$S (k, n) \cdot n!$
7	Distinct	Identical	No	Not allowed	None	$S (k, n)$
8	Distinct	Distinct	No	Not allowed	Any number of cells	$\sum_{i = 1}^{n} S (k, i) \cdot (i n) \cdot i!$
9	Distinct	Identical	No	Not allowed	Any number of cells	$\sum_{i = 1}^{n} S (k, i)$
10	Identical	Identical	No	Not allowed	None	$p_{n} (k)$ (partitions of $k$ into exactly $n$ positive integers)
11	Identical	Identical	No	Not allowed	Any number of cells	$p (k)$ (all integer partitions of $k$ )

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