How many different Huffman encoding for a given number of symbols
The answer is $C_{n-1} n!$ . That is, the $(n-1)$st Catalan number times $n$ factorial.

There are $C_{n-1}$ ways of making a complete binary tree with $n$ leaves, and there are $n!$ ways of assigning these leaves to the symbols to get a Huffman code.

This sequence goes 2, 12, 120, 1680, 30240, and is listed in the Online Encyclopedia of Integer Sequences as Quadruple Factorial Numbers, with a simpler formula than I give above.

The above argument shows that the quadruple factorial numbers give the number of ways of assigning $n$ symbols to codewords in a complete binary prefix code. It's not hard to show that you can assign probabilities to the symbols to make this encoding optimal (so it satisfies the definition of a Huffman code). It's not clear that a given Huffman coding algorithm will generate all of these, however.