Just how surjective is a cryptographic hash like SHA-1? From “Are common cryptographic hashes bijective when hashing a single block of the same size as the output” and “How is injective, inverse, surjective &amp; oneway related to cryptography”, it is suggested that cryptographic hashes are surjective. For avoidance of doubt, surjective means this:


  


whereby all the hash inputs (X) correspond to a reduced set of outputs (Y). This forms holes in the continuity of the output range, and we call them collisions.

Consider any hash function like SHA-1. The size of the possible input domain is $2^{160}$ if we stick to the block size. My linked answers suggest that the output co-domain is less than  $2^{160}$.

How much less exactly?

Are there any proofs or estimates to put a scale on this?  I wonder if the avalanche effect has any bearing on this?  This is probably extremely naive, but does anyone have anything better?
In this regard SHA-1 despite it's weaknesses can be viewed as a pseudo random function.
This means we are are throwing $n$ balls into $n$ bins.
An output bin remains empty if all the balls miss it.
Which happens with probability $(1-1/n)^n$ which is $1/e$ and that is the portion of output bins which are empty.
We also can estimate that the most populated bin has approximately $log(n)$ balls.