Suppose we have two finite sets $A, B.$ If we are to create a partial function $A\to B,$ for each $a\in A$ we have $|B|+1$ choices for the image of $a.$ Hence there must be $(|B|+1)^{|A|}$ partial functions between them.
Now, think of a partial bijection between them. In order to specify a such function we need two subsets with same cardinality (say $r$) from $A, B$ and a bijective map between those subsets. So, there must be $$\sum _{r=0}^{\min\{|A|, |B|\}}\dbinom{|A|}{r}\dbinom{|B|}{r}r!$$ number of such functions (?). My questions are,
a) Are these calculations correct?
b) Is there any closed form this summation? (at least when $|A|=|B|$)
What you have is correct. For $|A|=|B|$ the resulting sequence of numbers is OEIS A002720, with many references and several recurrences and generating functions but no closed form.