What is the maximal number of distinct values of the product of n permuted ordinals Because addition and multiplication of two order types are non-commutative operations, we have that for every integer n, given n ordinals, there are at most n! distinct possible values for the sum (or the product) of these n ordinals when all permutations are considered.
In fact, A. Wakulic (Fund. Math., XXXVI, pp 255-260) was able to establish the effective function F(n) giving the maximal possible number of distinct values for the SUM of n ordinals, when considering all permutations.
QUESTION: What is the effective function G(n) giving the maximal possible number of distinct values for the PRODUCT of n ordinals, when considering all permutations ?
Gérard Lang  
 A: For a lower-bound partial answer, note that there are at least as many ways to form a product as a sum, and so $F(n)\leq G(n)$. This is because we may replace an ordinal $\alpha$ with $\omega^\alpha$ and thereby turn multiplication into addition via $\omega^\alpha\cdot\omega^\beta=\omega^{\alpha+\beta}$. Since $\omega^\alpha=\omega^\xi$ just in case $\alpha=\xi$, this means that the ordinals $\alpha_1,\ldots,\alpha_n$ have exactly the same number of permuted sums as the ordinals $\omega^{\alpha_1},\ldots,\omega^{\alpha_n}$ have permuted products. 
Perhaps the functions for addition and multiplication are simply the same function? I don't see quite yet whether one can get more products by using ordinals not of the form $\omega^\alpha$. 
A: Simply take the ordinals $\omega+1,...,\omega+n$ and one obtains $n!$ distinct products (This solution was taken from Chapter 8 Problem 39 and Chapter 9 Problem 66 in the book Problems and Theorems is Classical Set Theory).
This follows from the easily provable fact that for natural numbers $r_{1},...,r_{n}$ we have $$(\omega+r_{1})\cdots(\omega+r_{n})=\omega^{n}+\omega^{n-1}r_{n}+\omega^{n-2}r_{n-1}+...+\omega r_{2}+r_{1}.$$
