Number of permutations of unspecified length

Question

We can see subsets as combinations of unspecified length of a set's elements, and there are, of course,

$$\sum_{k=0}^n {n \choose k}=2^n$$

subsets of a set with $n$ elements.

I was wondering whether there is a neat expression (analogous to $2^n$) for the number of permutations of unspecified length of the elements of a set.

For example, if we have five distinct items, the number of permutations for each possible length is given below; the total number of permutations (i.e., of unspecified length) would simply be the sum of these numbers.

      Length of permutation | Number of permutations
      ----------------------------------------------
          0                 |    1
          1                 |    5
          2                 |    5⋅4
          3                 |    5⋅4⋅3
          4                 |    5⋅4⋅3⋅2
          5                 |    5⋅4⋅3⋅2⋅1

Of course we can express the total number of permutations of unspecified length of $n$ items as $\displaystyle\sum_{k=0}^n {}_n\text P_k$ but is there is a neater expression that this sum evaluates to, or at least a conventional notation for this number?

Answer 1

Notice that your numbers can also be written, from the bottom row up, as

$$5!\left(1+1+\frac1{2!}+\frac1{3!}+\frac1{4!}+\frac1{5!}\right)$$

and this is the floor of $120e$