Generalizations of combinatorial quantities for cardinal numbers. Recently I've came up with an idea, that we can interpret $n!$ in terms of functions, as the amount of bijections of a certain set $X$, where $|X| =n$ (i. e. cardinality of the symmetric group on $X$) . So why not generalize this for $n$ being a non-finite cardinal number? As you can see in this post, assuming axiom of choice, $n! = 2^n$ when $n$ is infinite.
What are other generalizations of combinatorial quantities for general cardinal numbers? What is known about them?
 A: As Asaf mentioned in the comments, basically any combinatorial quantity that can be defined without using any finiteness assumption generalizes to infinite sets. 
A lot of the time it trivializes to infinite sets (at least in presence of AC) : as you noted $\kappa ! = 2^\kappa$, for any finite $n$, $|[\kappa]^n| = \kappa$ (where $[\kappa]^n$ is the set of $n$ element subsets of $\kappa$). 
However some may stay interesting : you can define, for any ordinal $\mu$ the set $[\kappa]^\mu$ of subsets of $\kappa$ of order type $\mu$ and this isn't as trivial as when $n$ is finite (a lot of Ramsey-type properties are defined with respect to this set); and one could define its cardinality as $\binom{\kappa}{\mu}$. Note that for this one, you can generalize it to arbitrary structures instead of cardinals/ordinals, and you get what some call structural Ramsey theory, which is really interesting. 
As Asaf also mentions, it isn't exactly clear what you're looking for. Probably any "combinatorial quantity" that you can define in terms of subsets and maps etc. can be quite straightforwardly generalized to the infinite case. Sometimes it trivializes or at least is computable in "known" terms (for instance we may declare that $2^\kappa$ is "known"), and sometimes it gets really hard and you enter a realm of hard (and interesting) infinite combinatorics, that are closely related to Ramsey theory, large cardinals and so on. It all gets a lot harder if you remove AC (or, depending on how you think of it, a lot easier : there's a lot less you can prove - but then you can prove independence statements etc.)
Of course, the question of whether it is computable also depends on what you consider to be known : for instance one might argue that $2^\kappa$ is not a "closed form" if that even makes sense.  
