How many different Huffman encoding for a given number of symbols In Huffman coding, if we have two symbols to be encoded, we will get the result either 01 or 10. If we have three symbols, we will get 12 different encoding. I am wondering if I give a arbitrary number of symbol, is there a formula to calculate how many different encoding we will have by Huffman coding?
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.